Party elders, conservative leaders and Republican colleagues have been pressuring Mr. Ryan, 45, who was Mr. Romney’s vice-presidential running mate in 2012, saying he should run for the good of the country, to rescue the institution of Congress and to protect the political interests of Republicans over all.

“There’s really only one choice: Paul Ryan is the guy to bring us together,” said Mr. Upton, of Michigan, the chairman of the Energy and Commerce Committee and a friend of Mr. Ryan’s, who said he had corralled him on the House floor.

Mr. Romney, in a statement, appealed to Mr. Ryan’s sense of duty. “I wouldn’t presume to tell Paul what to do, but I do know that he is a man of ideas who is driven to see them applied for the public good,” Mr. Romney said. “Every politician tries to convince people that they are that kind of leader; almost none are — Paul is.”

But even as Mr. Ryan was said to be reconsidering, his close associates cautioned that he had no intention of fighting for the job and would most likely accept it only by acclamation — including the support of the hard-line conservatives who pressed to oust Mr. Boehner and helped drive the majority leader, Kevin McCarthy of California, out of the speaker’s race on Thursday.

“If the conference decides as a whole that they want him to do it, it is possible,” said a Republican official close to Mr. Ryan who asked not to be identified because he was not authorized to speak publicly. “If there are still those guys who are holding out trying to make a point, it is not going to happen.”

While many hard-liners spoke admiringly of Mr. Ryan, they were also adamant that they would not simply jump to support handing him the speaker’s gavel. Many of them viewed Mr. Boehner’s resignation and Mr. McCarthy’s departure from the race as victories, and they have pledged to continue pressing for major changes in House rules to empower the rank-and-file members and weaken the leadership, including the speaker.