“I was encouraged that Steve Scalise this morning said that he thinks that after the election, Kevin McCarthy ought to be the person to replace me,” Mr. Ryan said. “I think that’s encouraging because what it shows you is that we have an intact leadership team that supports each other, that’s all heading in the right direction.”
Promises of continuity aside, questions remain over the shape and makeup of the Republican leadership. If President Trump, who is close to Mr. McCarthy, weighs in, his endorsement could seal the election for the majority leader. But until Election Day, members will not know which top leadership slot will open. If Republicans lose the majority, the next speaker will be a Democrat.
“It’s premature to be having these kinds of conversations, completely premature,” said Representative Charlie Dent, Republican of Pennsylvania, who is retiring. “If we retain the majority, that would be an argument for continuity of leadership. In the event we lose the majority, then members might say, ‘Hey, we need to go in a different direction with a clean slate.’ That’s why I think it’s very unhelpful for members to be jockeying for leadership right now.”
At the same time, some of his colleagues appear ready to show Mr. Ryan the door, despite his vow to “run out the tape.” Some House members, especially members of the hard-line Freedom Caucus, suggest it is highly unlikely that he will do so.
“We’re going to be running for re-election or election, and what’s the vision?” said Representative Scott Perry, Republican of Pennsylvania and a Freedom Caucus member. “When you make that investment of your vote, you want to know who is going to be embodying that vision. Who is that person right now? It’s not Paul Ryan.”
Whoever takes the helm will help determine the ideological direction of a conference that remains every bit as fractured as it was when Republicans drafted Mr. Ryan — a former vice-presidential nominee — to succeed John A. Boehner, who quit in the fall of 2015 rather than face a conservative rebellion.
Representative Matt Gaetz, Republican of Florida, who describes himself as a Freedom Caucus “sympathizer,” though not a member, said he expected a similar dynamic this time.
“Remember Paul Ryan didn’t become speaker because he ran for the job,” Mr. Gaetz said. “In our conference, running for speaker is a lot like running for pope. It’s not something where I expect vigorous zero-sum style campaigning.”
Still, many Republicans say that it is far from clear if a unifying figure will emerge — and that the dynamics will be different depending on whether the race is for speaker or for Republican leader. If the race is for speaker, which requires the votes of a majority of the entire House, the candidate will need the backing of the Freedom Caucus, which has about three dozen members.
But a race for majority leader, which simply requires the votes of a majority of the Republican conference, can be won without the Freedom Caucus’s support.
Representative Tom Rooney, Republican of Florida, who is retiring, said that the current situation, in which the Freedom Caucus “can basically sink any bill at any given time,” was untenable, and that whoever led the caucus would need to put an end to it.
“We can agree that the sky is blue and then at the last second, for some reason, it won’t be blue enough, and then we have to go to Pelosi to get votes,” said Mr. Rooney, referring to the House Democratic leader, Nancy Pelosi. “That can’t continue.”
A number of Republicans said they would be equally comfortable with Mr. McCarthy or Mr. Scalise. Mr. McCarthy is regarded as an affable presence in the House, a hard worker who is available to listen to members’ concerns.
When he sought the speakership in 2015, he was also damaged by a televised gaffe: He suggested on Fox News that the House committee investigating the 2012 attacks in Benghazi, Libya, was intended to damage Hillary Clinton’s presidential prospects. That clumsy moment did not inspire confidence in his ability to be the face of the House Republican conference, nor did an uncharitable assessment of the outgoing speaker — “I think John Boehner deserves a B-minus” — or a series of disjointed foreign policy interviews.
But Mr. McCarthy has rebounded thanks in part to his personal connection to Mr. Trump, who at a luncheon once called him “my Kevin.” Representative Tom Cole, Republican of Oklahoma, said Mr. McCarthy “is a better majority leader today than he was when he was seeking speaker.”
Mr. Scalise, who as the whip is in charge with rounding up votes to pass legislation, may be the sentimental favorite, given his struggle to overcome the grievous injuries he suffered when he was shot during a congressional baseball practice last year. His poignant story makes him a big draw on the fund-raising circuit.
“Nobody replaces Paul Ryan because he ran for vice president and he had some things going for him that nobody else did,” said Tom Davis, a Republican former congressman from Virginia.
“But I think people appreciate Steve Scalise and what he’s going through and suffered. You bring him to any Republican audience in the country, people will pay to see him and he’ll get a standing ovation,” Mr. Davis said, adding, “I think in McCarthy’s case they value his strategic mind and his ability to work things and to understand the system.”
For now, both Mr. Scalise and Mr. McCarthy are playing it coy. Hours after Mr. Ryan’s retirement announcement on Wednesday, Mr. McCarthy was mobbed in the Capitol by reporters asking his intentions. He waved them off.
“There is no leadership election,” he insisted. “Paul is the speaker right now.”