WASHINGTON — Vice President Mike Pence signaled support on Saturday for a futile Republican bid to overturn the election in Congress next week, after 11 Republican senators and senators-elect said that they would vote to reject President-elect Joseph R. Biden Jr.’s victory when the House and Senate meet to formally certify it.

The announcement by the senators — and Mr. Pence’s move to endorse it — reflected a groundswell among Republicans to defy the unambiguous results of the election and indulge President Trump’s attempts to remain in power with false claims of voting fraud.

Every state in the country has certified the election results after verifying their accuracy, many following postelection audits or hand counts. Judges across the country, and a Supreme Court with a conservative majority, have rejected nearly 60 attempts by Mr. Trump and his allies to challenge the results.

And neither Mr. Pence nor any of the senators who said they would vote to invalidate the election has made a specific allegation of fraud, instead offering vague suggestions that some wrongdoing might have occurred and asserting that many of their supporters believe that it has.

The senators’ opposition to certifying Mr. Biden’s election will not change the outcome. But it guarantees that what would normally be a perfunctory session on Capitol Hill on Wednesday to ratify the results of the presidential election will instead become a partisan brawl, in which Republicans amplify specious claims of widespread election rigging that have been debunked and dismissed for weeks even as Mr. Trump has stoked them.

The spectacle promises to set a caustic backdrop for Mr. Biden’s inauguration in the coming weeks and reflects the polarized politics on Capitol Hill that will be among his greatest challenges.

It will also pose a political dilemma for Republicans, forcing them to choose between accepting the results of a democratic election — even if it means angering supporters who dislike the outcome and could punish them at the polls — and joining their colleagues in displaying unflinching loyalty to Mr. Trump, who has demanded in increasingly angry fashion that they back his bid to cling to the presidency.

The conundrum is especially acute for Mr. Pence, who as president of the Senate has the task of presiding over Wednesday’s proceedings and declaring Mr. Biden the winner, but has his own future political aspirations to consider as well. On Friday, a federal judge dismissed a lawsuit brought by House Republicans to pressure Mr. Pence to do otherwise, and instead unilaterally overturn the results.

But on Saturday evening, Marc Short, his chief of staff, issued a statement saying that Mr. Pence “shares the concerns of millions of Americans about voter fraud and irregularities in the last election.”

The vice president, the statement continued, “welcomes the efforts of members of the House and Senate to use the authority they have under the law to raise objections and bring forward evidence before the Congress and the American people on Jan. 6th.”

In a joint statement on Saturday, the Senate Republicans — including seven senators and four who are to be sworn in on Sunday — called for a 10-day audit of election returns in “disputed states,” and said they would vote to reject the electors from those states until one was completed. They did not elaborate on which states.

The group is led by Senator Ted Cruz of Texas and includes Senators Ron Johnson of Wisconsin, James Lankford of Oklahoma, Steve Daines of Montana, John Kennedy of Louisiana, Marsha Blackburn of Tennessee and Mike Braun of Indiana, and Senators-elect Cynthia Lummis of Wyoming, Roger Marshall of Kansas, Bill Hagerty of Tennessee and Tommy Tuberville of Alabama.

Together with Senator Josh Hawley of Missouri, who announced this week that he would object to Congress’s certification of the election results, they bring to nearly one-quarter the proportion of Senate Republicans who have broken with their leaders to join the effort to invalidate Mr. Biden’s victory. In the House, where a band of conservatives has been plotting the last-ditch election objection for weeks, more than half of Republicans joined a failed lawsuit seeking to overturn the will of the voters, and more are expected to support the effort to challenge the results in Congress next week.

Representative Mo Brooks, Republican of Alabama, has said he will object to certifying the results, and with Mr. Hawley’s support, that challenge would hold weight, prompting senators and representatives to retreat to their chambers on opposite sides of the Capitol for a two-hour debate and then a vote on whether to disqualify a state’s votes. Both the Democratic-controlled House and Republican-controlled Senate would have to agree to toss out a state’s electoral votes — something that has not happened since the 19th century and is not expected to this time.

In their statement, the Republicans cited poll results showing most members of their party believe the election was “rigged,” an assertion that Mr. Trump has made for months, and which has been repeated in the right-wing news media and by many Republican members of Congress.

“A fair and credible audit — conducted expeditiously and completed well before Jan. 20 — would dramatically improve Americans’ faith in our electoral process and would significantly enhance the legitimacy of whoever becomes our next president,” they wrote. “We are acting not to thwart the democratic process, but rather to protect it.”

They also acknowledged that their effort was likely to be unsuccessful, given that any such challenge must be sustained by both the House, where Democrats hold the majority, and the Senate, where top Republicans including Senator Mitch McConnell of Kentucky, the majority leader, have tried to shut it down.

“We fully expect most if not all Democrats, and perhaps more than a few Republicans, to vote otherwise,” the senators wrote.

Senator Amy Klobuchar of Minnesota, the top Democrat on the Rules Committee with jurisdiction over federal elections, called the Republican effort a “publicity stunt” that would ultimately fail, but said it was dangerous nevertheless, amounting to “an attempt to subvert the will of the voters.” She noted in an interview that hundreds of millions of votes had already been “counted, recounted, litigated and state-certified” across the country.

“These baseless claims have already been examined and dismissed by Trump’s own attorney general, dozens of courts and election officials from both parties,” said Mike Gwin, a spokesman for Mr. Biden’s campaign.

While lawmakers have sought to register their opposition to past presidential election results by challenging Congress’s certification, the move has generally been more symbolic than substantive, given that the loser had already conceded and senators rarely joined with members of the House to force a vote. But as Mr. Trump continues to perpetuate the myth of widespread voter fraud, a growing number of Republicans in Congress have been eager to challenge the results, either out of devotion to the president or out of fear of enraging the base of their party that still reveres him even in defeat.

That is the case even though the vast majority of them just won elections in the very same balloting they are now claiming was fraudulently administered.

Mr. McConnell has discouraged senators from joining the House effort, and Senator John Thune of South Dakota, the No. 2 Republican, told reporters the challenge to the election results would fail in the Senate “like a shot dog,” prompting a Twitter rebuke from Mr. Trump.

Senator Ben Sasse, Republican of Nebraska, on Thursday condemned the attempt, calling it a “dangerous ploy” intended to “disenfranchise millions of Americans.” He accused fellow Republicans of making a political calculation to try to further their careers at the expense of the truth by tapping into Mr. Trump’s “populist base.”

But Josh Holmes, a Republican strategist and Mr. McConnell’s former chief of staff, warned that those involved in the effort would come to regret their stance.

“Rarely can you predict with 100% assurance that years from now everyone who went down this road will wish they had a mulligan,” Mr. Holmes wrote on Twitter.

Senator Patrick J. Toomey, a Pennsylvania Republican who has announced that he will not seek re-election in 2022, also blasted the effort, saying that Mr. Hawley, Mr. Cruz and others were “directly” undermining the “right of the people to elect their own leaders.”

For years, Mr. Trump has railed against contests in which he lost, disliked the outcome or feared he might be defeated. He objected to the results of the Emmys, falsely claimed President Barack Obama did not win the popular vote, asserted that Mr. Cruz “stole” a primary victory from him in Iowa in 2016 and predicted that the election in which he defeated the Democratic nominee Hillary Clinton would be “rigged.” In the months leading up to November’s election, he also warned that he would be cheated out of a victory, and refused to commit to a peaceful transfer of power.

As Mr. Biden racked up victories in November, Mr. Trump indulged in increasingly outlandish fictions, spreading disinformation about the election’s results and encouraging his followers to challenge the vote at every step. In recent weeks, as his legal defeats have stacked up, the president has become more vitriolic in his condemnations of Republicans who fail to support his false claims of having been the true victor in the election, and has lavished praise on those who parrot his accusations.

On Saturday, Mr. Trump cheered on the Republican senators who announced they would object to certifying the election, writing on Twitter: “Our country will love them for it!”

The vote tally and procedures in every battleground state that Mr. Trump contests have been affirmed through multiple postelection audits. Mr. Biden won the election with over seven million more votes than Mr. Trump and with 306 Electoral College votes, surpassing the threshold of 270 needed to win the presidency.

Nevertheless, more than a month after Mr. Biden’s victory, with increasing numbers in their party marching in lock step with Mr. Trump, some Republicans felt the need on Saturday to explain why they planned to vote to uphold the results of a democratic election.

“I swore an oath to support and defend the Constitution of the United States and that is what I will do Jan. 6,” Senator Lisa Murkowski of Alaska said in a statement. She is to face voters next November.

Senator Mitt Romney of Utah warned of the consequences of backing a bid to subvert the election’s outcome.

“I could never have imagined seeing these things in the greatest democracy in the world,” he said in a statement. “Has ambition so eclipsed principle?”

Maggie Haberman contributed reporting from New York, and Thomas Kaplan from Washington.