House Republicans will vote by secret ballot next Wednesday for the next speaker and will hold a final floor vote next Thursday.
The announcement came as the GOP conference met behind closed doors Wednesday morning to discuss the path forward for speaker and Congress's looming fiscal deadlines.
Rep. Paul Ryan, R-Wisconsin, told his colleagues on Tuesday night that he would "gladly serve" as speaker if the conference met several conditions and that the major GOP caucuses endorse him by this Friday.
"What I told the members is, if you can agree to these requests, and I can truly be a unifying figure, then I will gladly serve. And, if I am not unifying, that is fine as well. I will be happy to stay where I am, at the Ways and Means Committee," Ryan told reporters at a press conference after that meeting.
Ryan, however, faces an uphill battle convincing the 40-plus members of the House Freedom Caucus that he should be the next speaker.
The chairman of that conservative group, Rep. Jim Jordan, R-Ohio, told reporters on Wednesday that the Freedom Caucus hopes to meet with Ryan later in the day. A handful of members of the conservative group also met with him a day earlier, before he revealed his decision to run.
Rep. Raul Labrador, R-Idaho, another leader of the group, left open the possibility of supporting Ryan.
He said "it's fair to say" that a number of its members could support him, but added that "it's presumptuous to say that he's the only candidate."
The caucus had officially backed Rep. Daniel Webster, R-Florida, for speaker, when House Majority Leader Kevin McCarthy, R-California, and Oversight Chairman Jason Chaffetz, R-Utah, were still in the race.
On Tuesday night, Webster said he'll remain in the race.
Ryan's conditions included converting the GOP from being "opposition party" to a "proposition party," updating conference rules, unifying House Republicans and having time to spend with his family.
While Boehner has said he plans to resign from the speakership and from Congress on Oct. 30, he has also pledged to remain as speaker until his successor is chosen.
CBS News' Walt Cronkite contributed to this report.