The polls have closed in two runoff races that will determine the balance of power in the United States Senate and much of President-elect Joseph R. Biden Jr.’s ability to enact his agenda early in his term.

The races feature two Republican candidates — Senator Kelly Loeffler and David Perdue, whose Senate term ended on Sunday — trying to fend off their Democratic challengers, Jon Ossoff and the Rev. Raphael Warnock, in contests that have drawn national attention and unprecedented levels of campaign spending.

Both Mr. Biden and President Trump campaigned in the state on Monday, a sign of the high stakes of the races. If either Republican candidate wins, Senator Mitch McConnell, Republican of Kentucky, will remain the majority leader. But if both Democrats win on Tuesday, the party will gain control of the chamber, with Vice President-elect Kamala Harris serving as the tiebreaking vote.

Polling and early voting data — which showed that more than 3 million Georgians voted before Election Day — suggest it will be an extremely tight race. Polling is so close that in the FiveThirtyEight average, none of the four candidates broke 50 percent. A tight race could mean the counting could drag later into the week, as it did in the general election in November.

As votes are tallied, experts have told the public to brace for possible lead changes. Republicans are expected to jump to an early lead on Tuesday evening with in-person votes — which have favored that party during the coronavirus pandemic — counted faster, along with ballots cast in more rural and conservative areas. More Democratic counties in and around Atlanta are expected to take longer to tabulate their votes.

Mr. Biden narrowly carried Georgia in November, becoming the first Democratic presidential candidate to do so since 1992. That win has lifted the party’s hopes before the runoffs. But Democrats did not fare as well down ballot. Mr. Perdue far outpaced Mr. Ossoff by nearly 90,000 votes in November, suggesting that Democrats still have to make up ground to win the runoff.

The racial makeup of the final electorate will be crucial in a state where Black voters overwhelmingly support Democrats and white voters back Republicans. According to data compiled by georgiavotes.com, Black voters made up a larger share of early voters for the runoff — nearly 31 percent — than they did in the general election, when it was closer to 28 percent.

Mr. Warnock, who is the pastor of Ebenezer Baptist Church in Atlanta, the spiritual home of the Rev. Dr. Martin Luther King Jr., is seeking to become the first Black Democrat elected to the Senate from the South. He and Mr. Ossoff, a 33-year-old documentary film executive, ran in tandem throughout the runoffs.

Mr. Perdue, the former chief executive of Dollar General, and Ms. Loeffler, who was appointed to the Senate a year ago and is seeking a full term, have cast the race as a necessary check on Democratic power in Washington in 2021, though these efforts have been complicated by Mr. Trump’s continued insistence, without evidence, that he won re-election.

Mr. Trump’s repeated efforts to undermine the integrity of the presidential election have dampened Republican trust in the system, but party strategists still hope his rally on Monday in Dalton, in northwest Georgia, will lead to a surge of Republican voters at the polls.

President Trump falsely claimed on Tuesday that Vice President Mike Pence has the power to reject electors when the Electoral College vote is certified this week, continuing a pressure campaign the president has slowly increased in recent days.

“The Vice President has the power to reject fraudulently chosen electors,” Mr. Trump falsely claimed on Twitter.

As president of the Senate, Mr. Pence is expected to preside over the pro forma certification of the Electoral College vote count in front of a joint session of Congress on Wednesday. It is a constitutionally prescribed, televised moment in which Mr. Pence will name the winner of the 2020 presidential election, Joseph R. Biden Jr.

Mr. Trump has been trying for days to press the vice president to use his procedural role in the event as an opportunity to change the outcome of the election.

It is also a moment that some of Mr. Pence’s advisers have been bracing themselves for since Mr. Trump lost the election and stepped up his baseless claims of widespread voter fraud. During a rally in Georgia on Monday night, the president openly pressured Mr. Pence for the first time to satisfy his demand that the results be changed to benefit him.

Mr. Pence’s aides have said he will follow what the Constitution prescribes. But that Mr. Trump is now turning on an ally who has been among the most deferential to him over four years is a predictable final act of his presidency.

But no matter how distasteful it might be for Mr. Pence, J. Michael Luttig, a former United States Court of Appeals judge and leading conservative legal scholar, said that Mr. Pence had no choice but to simply count the votes.

“No president and no vice president would — or should — consider either event as a test of political loyalty,” Mr. Luttig said. “And if either did, he would have to understand that political loyalty must yield to constitutional obligation.”

House Republicans, with support from Mr. Trump, have also argued in court that Mr. Pence has the right to take matters into his own hands and eliminate electoral votes from any state that he chooses. But a federal judge, who was appointed by Mr. Trump, on Friday threw out a lawsuit that the Republicans brought to force the vice president to do so.

“The only responsibility and power of the vice president under the Constitution is to faithfully count the Electoral College votes as they have been cast,” Mr. Luttig said. “The Constitution does not empower the vice president to alter in any way the votes that have been cast, either by rejecting certain votes or otherwise.”

The effort by congressional Republicans to deny the presidential results found an echo in the Pennsylvania legislature on Tuesday, when Republicans voted not to seat a Democratic lawmaker who was elected in November and to remove the lieutenant governor, also a Democrat, as the presiding officer of the State Senate.

On a typically ceremonial day of swearing in members, Pennsylvania’s Senate majority refused to seat Senator Jim Brewster, whose narrow victory was officially certified but is being challenged in court.

In a contentious, chaotic session, Republicans also voted to remove Lt. Gov. John Fetterman as the Senate president and to replace him with the top Republican in the chamber.

The lieutenant governor refused at first to leave the rostrum, and for several minutes both he and the Republican voted into his place tried to recognize motions from the floor. Eventually, Mr. Fetterman stepped away.

“I was escorted out,” Mr. Fetterman said in an interview minutes later. “This was a corruption of the fundamental democratic franchise in our state.” He said Mr. Brewster’s win in November was certified by the secretary of the commonwealth and compared the state Republicans’ actions to President Trump’s efforts to subvert the outcome of his race.

Mr. Brewster, who has represented a region outside Pittsburgh for a decade, defeated Nicole Ziccarelli, a Republican, by 69 votes. She is challenging the results in federal court. At issue are several hundred mail ballots that did not have a handwritten date on their outer envelopes. Ms. Ziccarelli lost a challenge in state court.

Jake Corman, the president pro tem of the State Senate and a Republican, told reporters on Monday that his party believed it had to wait for the outcome of the legal challenge before filling the seat. “Our goal is to get it right, not get it fast,” he said.

But Democrats characterized it as a naked power grab. “This idea of having one party decide who is the real victor is a dangerous precedent we’re seeing played out on the national stage,” Mr. Fetterman said.

Jennifer Kocher, a spokeswoman for the Senate Republicans, accused Democrats of creating chaos. “Today, the order and decorum of the Senate were hijacked by Lt. Gov. John Fetterman and members of the Senate Democrat caucus, who failed to adhere to Senate rules,” she said.

The elections in Georgia won’t just determine the fate of the two Senate seats there and the balance of power on Capitol Hill. It will also reveal the extent to which President Trump has disrupted and damaged his own party.

For the past several weeks, Mr. Trump has instigated and intensified a battle in the Georgia Republican universe as he has sought to overturn his loss in the state and blame Republican leaders there for not helping him.

In response, the state’s Republicans have turned on one another, taking sides for or against the president as he continues in his obstinate — some say unlawful — effort to overturn the election results in Georgia, where he lost by nearly 12,000 votes.

The outcome of these Senate runoffs will show, on one level, how Republican voters have reacted to Mr. Trump’s quest to upend what he has falsely called a “rigged” election.

If Republicans ultimately do not turn out in large numbers, the blame will fall at least partly on the president for his efforts to raise doubts about the fairness of the election process.

The extent to which Mr. Trump is willing to go in that effort became fully apparent on Saturday, when he called Brad Raffensperger, Georgia’s Republican secretary of state and a Republican, urging him to “find” votes and recalculate the results of the state’s presidential contest in his favor, ignoring the official finding, already certified by the governor, that he had lost.

It was the culmination of efforts to overturn the election that began nearly two months ago. At every turn, Mr. Raffensperger and other Georgia election officials have debunked the conspiracy theories about voter fraud pushed by the president and his allies.

The short-term effect of Mr. Trump’s pressure campaign will become evident as the votes are counted in the runoffs pitting David Perdue and Kelly Loeffler, the two Republicans fighting to keep their Senate seats, against two Democrats, Jon Ossoff and the Rev. Raphael Warnock.

For Republicans in the state, the concern all along has been that Mr. Trump’s effort to undermine the election process will depress turnout in the runoffs, partly because he has stoked beliefs that the system itself is rigged and cannot be trusted.

Bill Crane, a Georgia political operative and commentator, said the president’s tactics, as well as the work of activists in the state who have claimed the general election was rigged, were tamping down Republican turnout. “Georgia is still conflicted about whether we should vote at all,” Mr. Crane said.

Data from early voting showed that the turnout in the runoff election was depressed in heavily Republican areas of the state, though analysts say that Republicans tend to favor voting on Election Day while Democrats are more likely to cast their ballots early.

President-elect Joseph R. Biden Jr. on Tuesday urged Georgians to vote and expressed continued optimism about unifying the nation, even as some Republicans in Congress push to overturn his election.

In an interview on WVEE-FM, an Atlanta radio station, Mr. Biden made a case for the importance of electing the Democratic candidates, Jon Ossoff and the Rev. Raphael Warnock, in the runoff elections on Tuesday for Georgia’s Senate seats.

“I need their votes in the Senate,” Mr. Biden said.

He said he was “feeling really optimistic about today,” and he made a simple request to Georgia residents: “Vote, vote, vote.”

Mr. Biden also made a pitch for Mr. Ossoff and Mr. Warnock in an interview with WFXE-FM in Columbus, Ga., declaring, “So much is at stake.”

The president-elect spoke a day after traveling to Atlanta for a drive-in rally with the two Democratic candidates. If both candidates win, their party will gain control of the Senate, with Vice President-elect Kamala Harris as a tiebreaking vote.

In the WVEE interview, Mr. Biden said their election would allow for the passage of $2,000 stimulus checks, and he suggested that the two Democrats could help provide support for his administration’s efforts to distribute the Covid-19 vaccine.

Mr. Biden said he envisioned establishing “thousands of federally run and federally supported community vaccination centers of various sizes across the country” in locations like high school gyms and N.F.L. stadiums.

And Mr. Biden, who ran for president with a message of bringing the country together and working with both parties, stuck to that theme despite plans among some Republicans in Congress to object to certifying the Electoral College results on Wednesday.

“There are enough really decent Republicans — you’re seeing them step up now in the United States Senate — who don’t want to be part of this Trump Republican Party,” Mr. Biden said, citing Senator Mitt Romney of Utah as one example. “There’s a whole bunch of them.”

ATLANTA — Georgia’s Election Day voters braved a bracing January chill on Tuesday , arriving at polling places to make their choices in two Senate runoff races that are among the most consequential in recent American history.

In liberal-leaning Atlanta, Whitney Leonard, 24, walked out of the West Hunter Street Baptist Church in a precinct in the West End neighborhood. Ms. Leonard said she voted for the Democratic challengers Jon Ossoff and the Rev. Raphael Warnock. But she said she was not beholden to the party.

Ms. Leonard said she felt that President Trump had proved himself immature and erratic, and she believed that Democrats taking control of the Senate was crucial to undoing the damage he had caused.

Before the presidential election in November, Ms. Leonard had never voted. Now, Ms. Leonard, who was previously incarcerated, said she was going to vote whenever the opportunity presented itself. “You don’t know how much of a privilege it is to vote until it’s been taken away from you,” she said.

In Dalton, the northwest city where Mr. Trump held a rally on Monday night, a steady flow of Georgians poured into Dalton State College to vote.

Northwest Georgia is a conservative stronghold, and Republicans knew their task was to overcome strong statewide Democratic turnout in early voting and absentee ballots. By Tuesday , Mr. Trump’s message to Republicans to get out and vote appeared to resonate.

“Turnout is high,” said Lane Lewis, 44, as he waited to enter the precinct. “You can tell because there’s never a line.”

The reliance on Election Day turnout is a risky proposition for Republicans, who must contend with Georgia’s changing population and growing urban areas that increasingly vote Democratic. It was also a forced choice — considering much of the Republican base is echoing Mr. Trump’s concerns about voter fraud in the presidential election — when it comes to absentee voting, and many have expressed similarly unfounded doubts about the Senate races.

Mr. Lewis said he waited until Election Day to vote because he trusted it would be counted then. He also said he believed Ms. Loeffler and Mr. Perdue would benefit from Republicans like him, who hold conservative values but have sometimes been put off by Mr. Trump’s language.

When asked if he thought Mr. Biden won Georgia in November, Mr. Lewis said, “I have doubts.”

Some have described their voting choices as a desire for balance or an aversion to having one party controlling two houses of government.

Joy Phenix, 55, voted for gridlock. “They need a backstop,” she said outside a polling place in the affluent Atlanta suburbs in east Cobb County, where a modest line of voters shuffled through all morning. Ms. Phenix said she voted for Mr. Perdue and Ms. Loeffler, but that if Mr. Trump had been elected, she would have voted for the Democrats.

Jasmine Knapp, a 30-year-old Dalton resident, said she was ready for the flood of campaign texts and television advertisements to end. Ms. Knapp, who declined to say who she supported, described herself as a conservative who voted Republican, but said she had not agreed with how some, like Mr. Trump, had claimed the election in November was rigged.

“You always hear every election cycle that this vote matters more than anything,” she said, “but that feels true this time.”

Local authorities in Washington are cautioning residents to avoid potentially violent agitators expected to gather downtown on Tuesday and Wednesday to amplify President Trump’s false claims of voter fraud in the November election.

The chief of the Metropolitan Police Department, Robert J. Contee III, said the police received information that people intend to show up to the demonstrations armed, a violation of local firearm laws. Mr. Trump is expected to appear at the rally on Wednesday and has encouraged his supporters to travel to Washington for the event. Some of his allies, including the conspiracy theorist and conservative radio host Alex Jones and some associates who recently received a pardon from the president, were expected to speak to crowds of Trump supporters and armed groups on Tuesday afternoon, one day before Congress begins the formal counting of the Electoral College votes.

A spokeswoman for the Eighty Percent Coalition, which was publicizing the event on Tuesday, did not return requests for comment.

On Monday the leader of the Proud Boys, a far-right group that has supported President Trump’s efforts to overturn the election results, was arrested on charges of destruction of property stemming from an episode in downtown Washington last month. Upon his arrest, Enrique Tarrio was found to have two high-capacity firearm magazines that bore the Proud Boys logo, and was charged accordingly with possession. Mr. Tarrio told detectives that he was delivering them to a buyer who had said he would also be in Washington this week, according to court papers.

Mr. Tarrio pleaded not guilty and was released by Judge Jonathan Pittman of the Superior Court of the District of Columbia. But he was ordered to stay out of Washington until his hearing this summer, and faces arrest if he tries to stay for the pro-Trump protests. A lawyer for Mr. Tarrio did not immediately respond to a request for comment.

“Protest organizations and the groups they represent have shown an alarming affinity for violence. Sadly, they have not been shy about suggesting the need for violence,” Marc Elrich, the executive of neighboring Montgomery County, Md., said in a statement warning local residents to avoid potential clashes between supporters of Mr. Trump and counterprotesters. “There is talk of disrupting the counting of votes in Congress, which would require extreme actions.”

Mayor Muriel Bowser of Washington on Monday requested support from the Army National Guard for the rallies. About 340 National Guard troops are expected to be present for the rallies and Customs and Border Protection has also placed tactical teams on standby to help protect federal property.

“We will not allow people to incite violence, intimidate our residents or cause destruction in our city,” Ms. Bowser said during a news conference on Monday. “We’re asking D.C. residents and people who live in the region to avoid confrontations with anybody who’s looking for a fight.”

There were a number of violent clashes last month between supporters of Mr. Trump and counterprotesters in the nation’s capital, where four people were stabbed.

When Georgia’s Senate runoff polls close tonight, elections officials will begin reporting three sets of vote totals: 2.1 million cast at in-person early-voting sites, 1 million cast by mail and those cast at Election Day precincts, a figure that officials estimate could be anywhere from 500,000 to 800,000.

There will be little consistency to how Georgia’s 159 counties report their results. Some will post all of their early in-person votes shortly after the polls have closed. But the mail ballots are likely to be slower. Most of these ballots have already been processed but not counted, meaning envelope signatures and addresses have been verified but the ballots have not been run through vote-counting machines.

Some counties will be faster than others to report their results. Fulton County, which includes Atlanta and has a reputation for being slow at reporting vote totals, is already far behind the state average in processing mail-in ballots.

Statewide, 74 percent of mail ballots have been processed. But Fulton County, a Democratic stronghold that is the largest county in the state, has processed just 66 percent of its ballots, according to the United States Election Project. In neighboring Cobb and Gwinnett Counties, suburbs that swung hard to President-elect Joseph R. Biden Jr., officials have processed 82 percent and 76 percent of their mail ballots — an indication that they will report results earlier in the night.

Other counties expected to be slow in reporting results include Henry County, a suburb south of Atlanta that has processed 57 percent of its mail ballots, along with the Atlanta suburbs of Clayton and Forsythe Counties; Chatham County, which includes Savannah; and Houston County, south of Macon.

One big unknown remains the size of Georgia’s Election Day turnout. The secretary of state’s office, which said on Tuesday afternoon that the average statewide wait time was one minute, is not reporting any turnout numbers until after the polls close. Officials with the Senate campaigns, political parties and outside groups working in the state had anecdotal data that showed a steady but not overwhelming turnout.

In the days after the Nov. 3 election, Georgia was one of several states where the vote count seemed to progress agonizingly slowly. But for several reasons, it’s unlikely that it will take as long to count the votes in today’s two runoff elections, which will determine control of the Senate. It’s even possible — but certainly not guaranteed — that we’ll know who won very early Wednesday.

Two factors work in favor of a faster count this time around. First, fewer races are on the ballot, which means less work for election officials. Second, after the general election, the Georgia State Election Board enacted a rule requiring counties to begin processing early and absentee ballots at least a week before future elections, allowing officials to complete time-consuming prep work even though they can’t actually count the ballots until the polls close.

If all goes smoothly, we could know who won by 1 a.m. Wednesday, according to David Worley, the sole Democratic member of the State Election Board.

The biggest questions are whether everything will, in fact, go smoothly, and just how close the races will be. Both races are very competitive, but in terms of when we’ll know who won, there’s a big difference between a race decided by two percentage points and one decided by 0.2 percentage points.

In an extremely tight race, results could be delayed several days while late-arriving ballots come in. The vast majority of Georgians have to get their ballots in by the time the polls close at 7 p.m. Eastern, but military and overseas voters have an extra three days as long as they mail their ballots by Tuesday. Then there are provisional ballots, which are cast on election day but take longer to process because officials have to verify each voter’s eligibility.

And, of course, there is always the possibility of a recount. Under Georgia law, a candidate can request one if the margin is less than half a percentage point. That wouldn’t be surprising: In the presidential race in November, Mr. Biden won by less than a quarter of a percentage point.

“Just like in November, it’s very possible Americans will go to bed without knowing who won,” The Associated Press said in guidance published Monday.

Two more Republican senators were making plans on Tuesday to object to electoral votes won by President-elect Joseph R. Biden Jr. on Wednesday when Congress meets to formalize his victory.

Senator Ted Cruz, Republican of Texas, plans to object to the certification of Arizona’s Democratic electors, according to a person familiar with his plans. And Senator Kelly Loeffler, Republican of Georgia, intends to object to the electors from her state, according to a person familiar with her thinking.

Mr. Cruz, a possible 2024 presidential candidate, is among 11 senators who have said in recent days that they would challenge the Electoral College results unless Congress agreed to create an independent commission to audit the results. But his earlier statements had been vague as to whether he would lodge a formal objection himself. His plan to object was first reported by The Washington Post.

His decision to do so, along with Ms. Loeffler’s, ensures that the House and Senate will formally debate whether to overturn the results in at least three states, prolonging what is normally a brief, ceremonial session by at least six hours and forcing at least three votes on whether to overturn Mr. Biden’s victory.

Senator Josh Hawley, Republican of Missouri, plans to object to Pennsylvania’s electors. Neither he nor Ms. Loeffler have ruled out lodging additional objections, and other Republican senators could still join the mix.

House Republicans are preparing to object to another three states — Michigan, Nevada and Wisconsin — but under law, they can only force a debate and vote on their challenges if a senator agrees to join them. None are expected to be successful. Majorities of both the House and Senate intend to vote to accept Mr. Biden’s victory despite intense pressure from President Trump to declare him the victor instead of Mr. Biden.

But by objecting, Mr. Cruz and the other Republicans are ensuring that Mr. Trump will get one final, high-stakes stand in the halls of Congress to argue his baseless claims of widespread election fraud. Senate Republican leaders fear it will fracture the party.

The person familiar with Mr. Cruz’s thinking, who requested anonymity, said he was not seeking to overturn the election, but to draw attention to his idea of forming an election audit commission. There is little chance that will happen, and every state in the country has already certified the results after verifying their accuracy, many following postelection audits or hand counts and recounts.

“We are going to vote to object to the electors — not to set aside the election, I don’t think that would actually be the right thing to do — but rather to press for the appointment of an electoral commission that can hear the claims of voter fraud, hear the evidence and make a determination as to what the facts are and the extent to which the law was complied with,” Mr. Cruz said Monday evening in an interview with the conservative radio host Mark Levin.

A lawyer advising President Trump in recent weeks has resigned from her law firm after it was revealed that she participated in the call where Mr. Trump pressured Georgia officials to help him reverse the state’s election results, the firm said in a statement on Tuesday.

The lawyer, Cleta Mitchell, has been advising Mr. Trump despite a policy of her firm, Foley & Lardner, that none of its lawyers should be representing clients involved in relitigating the presidential election.

“Cleta Mitchell has informed firm management of her decision to resign from Foley & Lardner effective immediately,” the firm said in a statement. “Ms. Mitchell concluded that her departure was in the firm’s best interests, as well as in her own personal best interests. We thank her for her contributions to the firm and wish her well.”

Foley & Lardner had begun to distance itself from Ms. Mitchell shortly after the call was first reported by The Washington Post on Sunday. As Mr. Trump has made increasingly specious claims about the election, he has been unable to attract high-profile, establishment lawyers to back his cause.

Ms. Mitchell was among several Trump aides who joined the call on Saturday in which the president threatened Georgia’s secretary of state, Brad Raffensperger, with “a criminal offense” if he failed to “find” enough votes to change the state’s presidential results.

Ms. Mitchell has been involved in representing far-right groups and conservatives for many years. She served on the board of the National Rifle Association and represented Tea Party groups that claimed they were illegally targeted by the I.R.S.

A federal judge in Atlanta on Tuesday denied a last-minute effort by President Trump to decertify Georgia’s election results, handing the president yet another courtroom loss just one day before Congress is scheduled to bring the presidential race to an official end.

The ruling from the bench by Judge Mark H. Cohen denying the emergency petition brought the number of legal defeats that Mr. Trump and his allies have suffered since Election Day to more than 60. The challenges have spanned eight states and dozens of courts, and have become more desperate as the vote in Congress on Wednesday to formally certify the victory of President-elect Joseph R. Biden Jr. has drawn closer.

In a complaint filed just hours before the start of the new year, Mr. Trump and his lawyers asked Judge Cohen to toss the verified results of Georgia’s presidential race, citing a litany of previously debunked fraud allegations. They claimed that officials in Georgia allowed dead people to vote, as well as unregistered voters, convicted felons still serving their sentences, and people who had registered to vote at post office boxes.

Mr. Trump raised many of these false accusations on Saturday in an hourlong phone call in which he pressured Georgia’s secretary of state, Brad Raffensperger, to help him “find” just enough votes in Georgia to win the election. On Monday, another Georgia state official, Gabriel Sterling, held a news conference rebutting nearly all of Mr. Trump’s false claims.

Judge Cohen denied the president’s emergency request at a brief hearing on Tuesday morning that journalists were blocked from covering remotely. While reporters have covered most of the hearings related to election challenges from Mr. Trump and his allies by either phone or video, Mr. Trump’s lawyers did not consent to allowing public access to the remote livestream of the hearing on Tuesday.

LONDON — President Trump has not said where he plans to go after leaving the White House on Jan. 20. But Scotland’s first minister made clear on Tuesday that he is not welcome there.

The official, Nicola Sturgeon, said that under newly imposed coronavirus restrictions, which prohibit all but essential travel, a visit by the president to one of his Scottish golf resorts, Trump Turnberry, would not be acceptable.

Rumors that Mr. Trump would head for Scotland flared after a Scottish paper reported that an American military version of a Boeing 757 — sometimes used by Mr. Trump — was scheduled to land at a nearby airport on Jan. 19, the day before Joseph R. Biden Jr. is to be sworn in as president.

“We are not allowing people to come into Scotland,” Ms. Sturgeon told reporters in Edinburgh, “and that would apply to him just as it applies to anybody else — and coming in to play golf is not what I would consider to be an essential purpose.”

A plain-spoken politician, Ms. Sturgeon said that she did not know what Mr. Trump’s travel plans were, but that she hoped his immediate plan was to exit the White House. On Monday, she imposed a lockdown on Scotland, which, like England, is battling a surge in coronavirus cases because of a rapidly spreading new variant.

Under the new rules, people are required to stay at home and to work from there when possible. Places of worship have been closed, and schools will operate by remote learning. Scotland has frequently moved faster and further than England to impose restrictions during the pandemic.

The White House initially declined to comment on the report, first published in The Sunday Post, but later denied it.

“This is not accurate,” the press secretary, Kayleigh McEnany, said on Tuesday. “President Trump has no plans to travel to Scotland.”

Two White House officials said that while there has been almost no concrete discussion of what Mr. Trump will do on Jan. 20 because he is so focused on trying to overturn the election results, they do not believe he is considering travel to Scotland.

Actually, it’s a trick question: Not a single major telephone pollster conducted a survey in Georgia in the month before the elections on Tuesday — partly out of exhaustion after the difficulties of 2020, and partly because of how dicey it always is to poll a runoff election, when turnout patterns become especially difficult to predict.

Only one of the 16 pollsters that have conducted surveys in these races uses the kind of peer-reviewed methods and live-interviewer phone polling that the nation’s top outfits tend to rely upon. (An earlier version of this briefing item said that none of the pollsters had used these types of methods. But one such poll was conducted in late November and early December on behalf of AARP.)

Since early December, the only public poll by a traditional, reputable firm that had been planned got called off in the middle of the process.

Still, based on what polling data is available, the averages suggest that the Democrats have a slight advantage.

Both of the Democrats — Rev. Raphael Warnock and Jon Ossoff — lead the Republican candidates — Kelly Loeffler and David Perdue — by roughly 2 percentage points in the polling averages calculated by FiveThirtyEight.

Republican strategists say their candidates are dealing from a position of strength, pointing to how the Republican candidates earned more votes than the Democratic ones in the general election in November.

Yet Democrats have some reason for cautious optimism, with over 3 million early votes already cast statewide. But, as the outcome in November showed, high turnout does not necessarily spell good fortune for Democrats.

Black voters, who lean overwhelmingly Democratic, appear to have made up a larger share of the early voting totals compared to November, according to data compiled by georgiavotes.com and by the U.S. Elections Project. As of Tuesday morning, 31 percent of early votes had been cast by Black voters, according to available data — up from roughly 28 percent in November.

Among the tens of thousands of Georgians who did not participate in the general election but have registered to vote since then, Black voters made up an outsize share, according to TargetSmart data published last week.

All told, the number of early ballots cast by Black voters has reached 85 percent of the total early votes cast by Black voters in the general election, compared to just 75 percent among white voters, who tend to vote Republican.