WASHINGTON — Senator Mitch McConnell has concluded that President Trump committed impeachable offenses and is pleased that Democrats are moving to impeach him, believing that it will make it easier to purge Mr. Trump from the party, according to people familiar with Mr. McConnell’s thinking.

The private assessment of Mr. McConnell, the most powerful Republican in Congress, emerged on the eve of a House vote to formally charge Mr. Trump with inciting violence against the country for his role in whipping up a mob of his supporters who stormed the Capitol while lawmakers met to formalize President-elect Joseph R. Biden Jr.’s victory.

In a sign that the dam could be breaking against Mr. Trump in a party that has long been unfailingly loyal to him, Representative Liz Cheney of Wyoming, the No. 3 Republican in the House, announced her intention to support the single charge of high crimes and misdemeanors, as other party leaders declined to formally lobby rank-and-file lawmakers to oppose it.

“The president of the United States summoned this mob, assembled the mob, and lit the flame of this attack,” Ms. Cheney said in a statement. “There has never been a greater betrayal by a president of the United States of his office and his oath to the Constitution.”

Even before Mr. McConnell’s position was known and Ms. Cheney had announced her plans, advisers to the Senate Republican leader had already privately speculated that a dozen Republican senators — and possibly more — could ultimately vote to convict Mr. Trump in a Senate trial that would follow his impeachment by the House. Seventeen Republicans would be needed to join Democrats in finding him guilty. After that, it would take a simple majority to disqualify Mr. Trump from ever again holding public office.

In the House, Representative Kevin McCarthy, the minority leader and one of Mr. Trump’s most steadfast allies in Congress, has asked other Republicans whether he ought to call on Mr. Trump to resign in the aftermath of last week’s riot at the Capitol, according to three Republican officials briefed on the conversations. While he has said he is personally opposed to impeachment, he and other party leaders did not mount an official effort to defeat the push, and Mr. McCarthy was working on Tuesday to build support for a censure resolution to rebuke the president for his actions.

Taken together, the stances of Congress’s two top Republicans — neither of whom has said publicly that Mr. Trump should resign or be impeached — reflected the politically fraught and fast-moving nature of the crisis the party faces. After four years of backing the president at nearly every turn and refusing to condemn even his most extreme behavior, party leaders were racing to distance themselves from a president many of them now regard as a political and constitutional threat.

Mr. McCarthy backed the electoral challenges Republicans lodged last week during Congress’s electoral count, voting twice to overturn Mr. Biden’s victory in key swing states even after the siege at the Capitol. Mr. McConnell had broken with Mr. Trump just as the rioters were breaching the building, warning of a descent into a “death spiral” for democracy if the efforts were to prevail.

Mr. Trump has shown no trace of contrition. On Tuesday, in his first public appearance since the siege of the Capitol, he told reporters that his remarks to supporters at a rally that day — in which he exhorted them to go to the Capitol and “fight” so Republicans would reject the election results — had been “totally appropriate.” It was the specter of his impeachment, he said, that was “causing tremendous anger.” But with Twitter having suspended his account for good, Mr. Trump no longer has his favorite weapon to train on lawmakers who cross him, which could curtail the blowback they face for voting against him.

His party’s rapid turn against Mr. Trump unfolded as the House met into the night on Tuesday to debate and vote on a resolution formally calling on Vice President Mike Pence to invoke the 25th Amendment to strip the president of his powers, a move that Mr. Pence has indicated he was disinclined to make.

“We’re trying to tell him that the time of a 25th Amendment emergency has arrived,” Representative Jamie Raskin of Maryland, the author of the resolution, said before the vote. “It has come to our doorstep. It has invaded our chamber.”

Anticipating Mr. Pence’s refusal to act, Democrats planned a Wednesday vote on a single article of impeachment charging Mr. Trump with “inciting violence against the government of the United States.”

The White House expected roughly two dozen Republicans to support the charge, according to a senior administration official who insisted on anonymity to share a private assessment. Along with Ms. Cheney, Representatives John Katko of New York and Adam Kinzinger of Illinois announced they would support the charge. Just over a year ago, House Republicans rallied unanimously against Democrats’ first impeachment of Mr. Trump.

Forgoing a lengthy investigation, Democrats released a 76-page report collecting public information about the attack — including social media posts, news articles and other statements — and laying out a legal justification for impeachment.

“It is true that the president’s remaining term is limited — but a president capable of fomenting a violent insurrection in the Capitol is capable of greater dangers still,” they wrote. “He must be removed from office as swiftly as the Constitution allows. He must also be disqualified to prevent the recurrence of the extraordinary threat he presents.”

Mr. McConnell has indicated he wants to see the specific article of impeachment that the House is set to approve on Wednesday, and to hear the eventual arguments in the Senate. But the Senate Republican leader has made clear in private discussions that he believes now is the moment to move on from Mr. Trump, whom he blames for causing Republicans to lose the Senate. Mr. McConnell has not spoken to Mr. Trump since mid-December, when the senator informed the president he would be recognizing Mr. Biden as president-elect after the Electoral College certified it.

David Popp, a spokesman for Mr. McConnell, declined to comment on Tuesday, instead pointing a reporter to a speech the Kentucky Republican made when he returned to the Senate floor after Wednesday’s siege.

“This failed attempt to obstruct the Congress, this failed insurrection, only underscores how crucial the task before us is for our republic,” Mr. McConnell said as the Senate reconvened to complete the electoral count disrupted by the mob. “Our nation was founded precisely so that the free choice of the American people is what shapes our self-government and determines the destiny of our nation.”

On Monday, Mr. Biden telephoned Mr. McConnell to ask whether it would be possible to set up a dual track that would allow the Senate to confirm Mr. Biden’s cabinet nominees and hold a Senate trial at the same time, according to officials briefed on the conversation who disclosed it on the condition of anonymity. Far from avoiding the topic of impeaching Mr. Trump, Mr. McConnell said it was a question for the Senate parliamentarian, and promised Mr. Biden a quick answer.

After whipping votes to ensure Mr. Trump was not found guilty in the impeachment trial last year, Mr. McConnell, has turned sharply against Mr. Trump. Last week, in a memo to Senate Republicans, he indicated it would be difficult to hold a trial before Jan. 20, but notably did not defend the president.

Senator Chuck Schumer of New York, the Democratic leader, called on Mr. McConnell to use emergency powers to call the Senate back for a trial as soon as the articles were adopted.

“The bottom line is that Leader McConnell has the ability to call us back into session and we can then move to convict Donald Trump, draw on the impeachment trial and try him,” Mr. Schumer told reporters in New York. “And that’s what we hope McConnell will do.”

But because the Senate is in recess, the two leaders must agree to do so or else a trial would begin no sooner than Jan. 19, when they return. The next day, with Mr. Biden’s inauguration, Democrats will take operational control of the Senate, where they will have a working majority by dint of Vice President-elect Kamala Harris’s power to cast tiebreaking votes.

For Mr. McConnell and other Republicans, the crisis offered an opportunity to bar Mr. Trump from seeking the presidency again in 2024, as he has repeatedly mused with allies about doing.

“Congressional Republicans must evaluate this latest Trump situation and look at the best long-term solutions for the country,” said Scott Reed, a longtime Republican strategist. “This is now totally about Trump, not his supporters, and a permanent purge must be on the table.”

But that prospect has created a conundrum for Republicans who, understanding the deep affection for Mr. Trump among a powerful segment of their party’s core supporters, are concerned they could pay a steep political price for abandoning him.

In the days since the attack, Mr. McCarthy has veered from asking Republican colleagues if he should call on Mr. Trump to resign to privately floating impeachment to his current posture, opposed to impeachment but open to a censure. After he and over 100 other House Republicans opposed the certification of the Electoral College, Mr. McCarthy is now finding anger and regret among his Republican colleagues and is moving to take a tougher line with the president.

Reports emerged Monday from Axios that the House Republican leader had had an intense conversation with Mr. Trump, during which the president floated conspiracy theories about the rioters and Mr. McCarthy pushed back forcefully.

Unlike Mr. McCarthy, Mr. McConnell strongly opposed the effort by Senators Josh Hawley of Missouri and Ted Cruz of Texas to object to electoral votes from certain states.

The two senators have received a hefty share of criticism from across the ideological spectrum, but there has been fallout for other Republicans who joined their ranks as well.

A number Republican lawmakers and aides were worried that Senator Rick Scott of Florida, who is taking over the party’s Senate campaign arm, would find it highly difficult to raise money with corporate America moving to freeze out Republicans who refused to certify the Electoral College. Americans for Prosperity and its political action committee, funded by the influential conservative Koch network, will evaluate future support of politicians based on their actions last week, its chief executive told The Wall Street Journal.

Mr. Biden has made clear, in public and private, that he will not oppose the Democratic push to impeach Mr. Trump, even though his advisers and some lawmakers in his party are concerned about the impact it could have on his first days in office.

When he spoke with Mr. McConnell about the matter, the Senate leader left Mr. Biden with a bit of welcome news.

Mr. McConnell, who led the 2016 blockade against confirming Judge Merrick B. Garland when he was President Barack Obama’s nominee for the Supreme Court, told Mr. Biden that he would vote to confirm Judge Garland as attorney general.

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