Ryan informed several members on the House floor of his deliberations, Rep. Chris Stewart, R-Utah, told CNN. Stewart added that he urged Ryan to run, with the Wisconsin Republican replaying that he was "thinking and praying on it."
"Paul's looking at it but it's his decision," McCarthy said on CNN. "If he decides to do it, he'll be an amazing speaker but he's got to decide on his own."
Rep. Darrell Issa, R-California, said that Ryan told him privately he is thinking about it as well. "I think he's gone from a 'hard no' to he knows he has to consider it," he said.
Ryan, the Ways and Means Committee chairman and Mitt Romney's vice presidential running mate in 2012 is respected throughout the conference and on paper could bridge the gap between tea party-linked conservatives and more establishment Republicans.
Romney even called Ryan and asked him to run, an aide to the former presidential candidate said.
"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," Romney said in a statement. "Every politician tries to convince people that they are that kind of leader; almost none are -- Paul is. Paul has a driving passion to get America back on a path of growth and opportunity. With Paul, it's not just words, it's in his heart and soul."
There may be no way for Ryan to avoid the calls to run.
"He needs to do this for the team," said Energy and Commerce Committee Chairman Fred Upton, R-Michigan.
"There's going to be a lot of pressure on Paul," said Rep. David Jolly, R-Florida. "If Paul really means no, he better keep his phone off the next two weeks. Or throw it in Lake Michigan."
Ryan spokesman Brendan Buck issued a statement following Friday morning's GOP conference meeting, that did not rule out a run. "Chairman Ryan appreciates the support he's getting from his colleagues but is still not running for speaker," Buck said.
Leaving Capitol Hill Friday afternoon, Ryan declined to comment on his decision.
"Right now I'm going to make my flight so I can make it home for dinner," he told reporters. "Sorry guys I'm just going to go. The Packers are at home. They're going to beat the Rams and cover the spread."
Boehner's plea: work together
Boehner indicated to his fellow Republicans on Friday morning he will stay as long as possible in the speaker's job, although he still wants to "be out of here by the end of the month," Rep. Dennis Ross said, adding that Boehner urged members to "hang in there."
The speaker pled with his fellow Republicans to work together to break the impasse.
"While we go through this process, we've got to continue to address the people's priorities. This institution cannot grind to a halt," Boehner said in the meeting, according to a source in the room.
Several high-profile issues face Congress and the White House later this year, including the need to address the debt limit in early November and the federal government budget by December 11.
Boehner urged GOP lawmakers not to put up barriers between each other. "Don't start erecting walls between us as members. We all came here to help advance the conservative cause," he said.
"It's not helpful for one group of members to say they will only vote for this candidate on the floor," he added. "And it's not helpful for another group of members to say they will only vote for different candidate on the floor. It's up to the people in this room to listen to each other, come together, and figure this out."
He later said: "Time for us to take the walls down, open up our ears and listen to each other."
And addressing rumors that he might step down as majority leader, McCarthy told his fellow Republicans he plans to remain in the job, a source in the room said.
The meeting was centered on the rules and process going forward, members said.
"We are doing what needs to be done, said Rep. Scott Rigell, R-Virginia. "And Kevin (McCarthy) spoke very eloquently this morning and said, 'if this causes us to do what we need to do in terms of reforming the rules and the House rules. Then that's a good thing."
Can Ryan unite the caucus?
As the surprise of McCarthy's decision not to seek the top job wore off Thursday afternoon, eyes turned to Ryan, the former vice presidential nominee and chairman of the Ways and Means Committee. Influential and well-respected throughout the GOP caucus, Ryan could be a peacemaker between the warring factions.
Deputy Majority Whip Tom Cole, R-Oklahoma, also wants Ryan in the job.
"I think eventually Paul Ryan will get into this thing, I really do," Cole said Friday on MSNBC's "Morning Joe." "I think we've had two guys fall on their swords for the good of the conference. In this case it's hard not to step up when there's an overwhelming demand."
And for good measure, he added an appeal to Ryan's sense of duty.
"Paul's got a strong sense of doing the right thing," Cole said. "I'm just so confident he'll make the right choice."
Boehner and Ryan have a long relationship: Ryan volunteered for Boehner's campaign when he was a student at Miami University in Ohio in the 1990s, before launching his own political career. But Boehner has not publicly called on Ryan to replace him.
On paper, Ryan is a strong candidate. But his backing among the current House leadership could be a drag on his candidacy, as the conservative backlash against Boehner's team shows no sign of abating.
Conservatives do praise many of Ryan's policy positions, yet are already warning Ryan would need to agree to a series of demands to change House rules and agree to their policy agenda.
Rep. Charlie Dent, a moderate and critic of the Freedom Caucus, said more needs to change than just the man or woman in the chair.
"At the end of the day, it's not who we put in that job," Dent said on CNN. "If we don't change the political dynamic, the next speaker will suffer the fate that John Boehner did."
"We must assemble bipartisan coalitions to pass any meaningful legislation," Dent added. "That's the way this place has been operating. We have to accept that reality and move forward."
Rep. Mimi Walters, a California Republican, called for compromise in selecting the next speaker. "I think we all need to give a little bit," she said on CNN. "We're never going to find the perfect person because nobody is perfect but what we have to remember is we have to put the American people first."
Democratic Rep. Luis Gutierrez on MSNBC called Ryan one of the smartest men in the GOP.
"He would be good for the country," Gutierrez said. "He would be good for the Republican Party. Paul Ryan is the kind of individual that would work with people on the other side of the aisle and that's what we need.'"
Chaffetz won't run if Ryan does
Friday morning, Rep. Jason Chaffetz, the Utah Republican whose candidacy helped draw people away from McCarthy, said he would drop out of the race if Ryan runs, a sign conservatives are willing to rally behind the Wisconsin Republican.
"He's certainly in my mind the most qualified person to do it and I hope he'll do it," Chaffetz said on MSNBC.
Other names that Cole said could be viable if Ryan stands firm and doesn't run include Minnesota Rep. John Kline and Georgia Reps. Lynn Westmoreland and Tom Price. Westmoreland told CNN he was considering it. And Cole's own name has been floated by his peers -- something he said Thursday he wouldn't rule out, though he said there were "better" options than himself.
And Rep. Daniel Webster, R-Florida, who won the Freedom Caucus' endorsement, also remains a candidate. Webster Friday said he is still running but left open the possibility of abandoning his bid if he does not win support in conference.