Ryan laid out his guidelines in a closed-door session with the full House GOP conference after weeks of lobbying from senior members of his party for him to take on the difficult task of leading the divided conference -- a job Ryan repeatedly said he has never wanted.
"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. And I think our country is in desperate need of leadership," Ryan told reporters after addressing his colleagues.
Ryan will run only if he's supported by three groups inside the House Republican conference: the House Freedom Caucus, the Republican Study Committee and the moderate Tuesday Group, his spokesman, Brendan Buck, said.
Ryan also had a few more demands: He wants changes to House rules made as a team -- a major demand of the House Freedom Caucus; he wants to make it harder to overthrow a sitting speaker; and he wants a better work-life balance than Boehner had.
He also emphasized the importance of unified support for the next speaker.
Ryan told his colleagues he is willing to take "arrows in the chest but not in the back," a GOP source inside the meeting told CNN.
Should the conference agree to his stipulations, "I am happy and willing to get to work."
The pressure to run
The meeting Tuesday night capped days of hand-wringing among House Republicans who recruited Ryan after House Majority Leader Kevin McCarthy abruptly dropped out of the leadership race.
The GOP source said Ryan told colleagues he "does not want to be the third log in the fire," referring to Boehner and McCarthy.
The conditional offer by Ryan ended a day full of back-channel meetings as the anticipation for his decision reached a fever pitch.
Boehner and the top voices in the conference leadership all pressed Ryan to get in, calling him the only choice for the party. The former vice presidential nominee softened his position as his close friends and colleagues besieged him as the party's only hope.
Ryan spent last week's House recess in Wisconsin with his family, mulling the decision. His young children are a substantial factor in his reticence to take on the demanding job.
But, he said Tuesday, they're also an influence in going after it.
"My greatest worry is the consequence of not stepping up, of some day having my own kids ask me, 'When the stakes were so high, why didn't you do all you could? Why didn't you stand and fight for my future when you had the chance?'" Ryan said to reporters.
His remarks in closed session Tuesday pleased his supporters, and some of the conservatives he'll have to convince to support his bid didn't rule out their support.
"I thought he laid out a very positive vision. It was quintessential Paul Ryan, talking about how we can lead, be on offense," said Oregon moderate Republican Rep. Greg Walden. "He gave a compelling speech with vision about how to move forward and I think you could probably hear the standing ovation applause."
Indeed, raucous applause was audible by reporters gathered in the adjoining hallway to the meeting.
House Freedom Caucus member Rep. Matt Salmon, R-Arizona, said he still needs to talk to Ryan before deciding whether to support his bid. The conservative caucus had endorsed Rep. Daniel Webster, R-Florida, for speaker and not yet moved away from that endorsement.
But Salmon also said he doesn't think it's "an impossible task" for Ryan to win their backing.
And just before Ryan met with his colleagues in full, Boehner gave him an extra push -- saying definitively on Fox News' "Special Report" that Ryan would announce his decision Tuesday night.
"I hope he does decide to run, and if he does, I think he'll be elected," Boehner said.
The comments pushed Ryan's spokesman to tweet
that Ryan would not be making a final decisions Tuesday night.
If Ryan ultimately feels that he can't unite the party and passes on the job, Republicans are growing nervous that there is no one else who could.
"It worries me a lot," Sen. John McCain, R-Arizona, told CNN. "Because a leaderless House of Representatives is not something that is conducive to legislative successes -- and it diminishes us even more in the esteem of the American people."
Conservatives in the House Freedom Caucus signaled Tuesday that Ryan will need to work to get its support, something the Wisconsin Republican has shown little interest in doing.
The caucus tweeted before he spoke that "The next speaker must follow House rules and commit to an open process for debating and amending legislation. Let the House work its will."
"I have concerns with anybody who is not going to give us a reform agenda," said Rep. Raul Labrador, R-Idaho. "If anybody thinks we are just going to get behind somebody just because they have a national name, they are sadly mistaken."
Alabama GOP Rep. Mo Brooks, also a member of the House Freedom Caucus, said he would hold a tele-town hall with his constituents to discuss whether they would support Ryan as speaker. But Brooks cited Ryan's past support for immigration reforms as a reason he was already skeptical.
"There is growing concern in my district, as more and more people become aware of Paul Ryan's amnesty and open borders immigration position and that concerned deepened when Luis Gutierrez endorsed him for speaker," Brooks said.
Gutierrez -- who has praised Ryan -- is one of the most vocal House Democrats pushing for comprehensive immigration reform.