Rep. Paul Ryan (R-Wis.) announced that he would run for House Speaker only if his set of demands were met. Washington Post National Political reporter Robert Costa explains what these conditions are and what's in store for Ryan next. (Alice Li/The Washington Post)

Hard-line conservatives cleared a path Wednesday for Rep. Paul Ryan to become House speaker when some of his most disgruntled fellow Republicans signaled that they would support his bid for the top job.

The decision to back Ryan by the House Freedom Caucus, a group of nearly 40 lawmakers that has risen in power and stature since its founding this year, came after the Ways and Means Committee chairman spent much of his day courting its support.

The group stopped short of an official endorsement, which would have required 80 percent support, but members said a “supermajority” of the caucus would back a Ryan bid for speaker. Ryan set out a series of conditions Tuesday under which he would consider seeking the speakership; the most challenging of those was unity among all of the House Republican Conference’s warring factions. The support of the Freedom Caucus was regarded as one of the huge obstacles to meeting that condition.

In a statement, Ryan said he did not view the lack of a formal endorsement as a rejection: “I believe this is a positive step toward a unified Republican team.”

Rep. Raúl R. Labrador (R-Idaho), a co-founder of the Freedom Caucus, emerged from a lengthy evening meeting Wednesday and said there was “consensus that we need to move forward because it’s time for the conference to unite.”

“It’s time for everybody to work together and make the Republican Party stronger,” he said. “That’s what we’re trying to do even with the reservations that some people have about Paul Ryan being speaker.”

Ryan could still decide not to serve as speaker, and some conservative activists have engaged in a vigorous campaign to cast doubt on his record, which might give some members cold feet before votes are cast next week.

But the level of Freedom Caucus support represents the first thaw in the increasingly frosty relationship between tea party conservatives and establishment Republicans. It also paves the way for fresh GOP leadership heading into imminent clashes with President Obama over the national debt and federal spending.

The Freedom Caucus met with Ryan for an hour in the Capitol earlier in the day. Many of its members had balked at the conditions Ryan attached to his decision to serve as speaker, and the meeting represented their first chance to question him directly about his plans.

The meeting broke up without resolution, setting up a high-stakes decision for a group that played a key role in easing the current speaker, John A. Boehner, into retirement and blocking Majority Leader Kevin McCarthy’s bid to succeed him.

The group reconvened in the evening to debate whether to abandon their previous endorsement of Rep. Daniel Webster (R-Fla.) in favor of Ryan, who signaled Tuesday that a Freedom Caucus endorsement would be a prerequisite to him agreeing to serve as speaker.

Rep. Paul Ryan announced on Oct. 20 that he will run for House speaker, saying "this is not a job I ever sought." (C-SPAN)

Ryan’s declaration that he would serve as House speaker if and only if he receives formal backing from major House GOP factions — including the Freedom Caucus — effectively gave the group veto power over his ascent. It also exposed fissures in the typically close-knit caucus.

Some, citing Ryan’s demand to jettison the House rule allowing a simple majority to oust a speaker at virtually any time, said it would be nearly impossible for him to earn their support. Others argued that Ryan could be the type of transformative leader that House Republicans need.

The Freedom Caucus was also facing the prospect of further alienating the rest of the House GOP, and a potentially crippling loss of credibility, if it were to reject Ryan. Many mainstream conservatives saw Ryan as the best chance, maybe the only chance, to unite their fractious party.

“It would be an embarrassment to them” if the Freedom Caucus dismissed Ryan, said Rep. John Kline (R-Minn.), chairman of the Education and Workforce Committee. “What he’s doing is selfless.”

Ryan appeared to calm some nerves in the afternoon meeting, making clear that he did not want to end the ability of the House membership to remove the speaker — only change the procedures for doing so. He also gave reassurances that he would respect the “Hastert Rule” — the informal practice of former speaker J. Dennis Hastert that required the majority support of the Republican conference before moving legislation to the House floor.

Rep. Mo Brooks (R-Ala.) said Ryan agreed that legislation should be moved only with the support of the majority of the majority. In particular, he said, that calmed immigration hawks wary of Ryan’s past support for measures that would offer illegal immigrants a path to citizenship.

But Ryan faced a big challenge in getting the group’s endorsement, Brooks said: “Paul Ryan probably made some progress . . . but to get 80 percent of Freedom Caucus to switch from Daniel Webster?” he said. “It’s going to be difficult for Paul Ryan to shift that in two, three days.”

Ryan said little upon departing the afternoon meeting, calling the gathering an “exchange of ideas on how to make Congress work better.”

Earlier in the day, Boehner announced that Republicans will vote internally to nominate a speaker next Wednesday, with a floor vote to follow Thursday. The announcement was made after Ryan said Tuesday night that he would run for speaker only if his terms were met.

“This is not a job I’ve ever wanted, I’ve ever sought,” Ryan said. “I came to the conclusion that this is a very dire moment, not just for Congress, not just for the Republican Party, but for our country.”

Those demands include not only the endorsements and the rule changes, but also giving Ryan time with his young family.

By the time the meeting wrapped up Wednesday night, only a “small handful” of members had reservations, said Rep. Mick Mulvaney (R-S.C.), signaling that Ryan had secured the 218 House votes necessary to prevail in the floor vote.

“I think he satisfied many of us that he was going to change business as usual in Washington, D.C.,” Mulvaney said.

Another sticking point for Freedom Caucus members was their endorsement of Webster, which was made earlier this month and played a role in ending McCarthy’s bid for the speakership.

Webster’s focus on procedural reforms, honed during his years as speaker of the Florida House of Representatives, has won him an avid following among hard-liners who feel marginalized by the GOP’s establishment.

“You’ve got a bunch of alpha people in here,” said Rep. Paul A. Gosar (R-Ariz.), a Webster supporter. “You don’t need another alpha leader.”

Webster made clear Wednesday that he would not stand aside for Ryan: “People are responding to what I’m saying. They’re sick of how this place is run, of the dog-and-pony shows on committees. They want a return to bills from members being considered, rather than approving the leadership’s bills.”

But Ryan’s near-bulletproof reputation among conservatives as a visionary and policy expert allowed hard-liners to look past his 16-year congressional tenure and trust him in a way that they never trusted Boehner or McCarthy.

“We’ve got a little way to go,” Rep. Marlin A. Stutzman (R-Ind.) said during the Freedom Caucus deliberations. “But I’m willing to start those conversations because I trust Paul. He’s earned my trust. I’m willing to keep talking.”

Two other major GOP caucuses have yet to weigh in, but neither is seen as an obstacle to Ryan.

Ryan met Wednesday with the Republican Study Committee, a more mainstream conservative group that counts more than two-thirds of the Republican conference as members. Its members were surveyed by secret ballot Wednesday, and its steering committee is expected to decide Thursday whether to grant an endorsement.

Also Thursday, Ryan is set to meet with the centrist Tuesday Group, which is expected to embrace his speaker bid.

Karoun Demirjian, Kelsey Snell and David Weigel contributed to this report.