The subsidies, estimated at $7 billion this year, were challenged in federal court by the House in 2014 and found unconstitutional because Congress had not approved them. Mr. Trump announced last week that he was cutting them off and Mr. Alexander said he feared millions of people could now find insurance out of their financial reach.
Despite the serious negotiations, there was no firm commitment that a compromise would reach the Senate floor if one could be struck. In fact, Mr. Alexander deferred to Mr. McConnell last month when Mr. McConnell wanted to take one last shot at overturning the health care law. He broke off his talks with Ms. Murray, resuming them only when the repeal effort faltered once again.
All sides agree that Mr. Trump — in conversations with Mr. Alexander and Senator Chuck Schumer of New York, the Democratic leader — strongly encouraged the bipartisan talks and initially seemed to embrace the agreement to restore the subsidies as a way to buy time for another repeal push. So backers of the plan were blindsided when the president said on Twitter on Wednesday that he could not back what he considered an insurance industry bailout.
Mr. Schumer said the president was undercutting himself by wavering on deals he had helped set in motion, not just on health care but also on issues like immigration. Mr. Schumer suggested that Mr. Trump caved to conservative opposition to the health care compromise.
“This president cannot govern if whenever the hard right frightens him and says, ‘Jump,’ he says, ‘How high?’” Mr. Schumer said.
Backers of the compromise face a tough task in selling the idea to many Republicans who would now be voting not to repeal the health care law, but to temporarily prop it up to avoid a breakdown in the marketplace. In addition, many on Capitol Hill say the compromise could be an important first step toward stabilizing the overall program, making future attempts to repeal it even more difficult.
Mr. Trump’s view will be crucial. His endorsement could provide important political cover for those nervous about voting to bolster the health care law while his opposition could cause Republicans to flee, fearing they will come under attack from the president and his allies.
Mr. Alexander is not retreating. Trying to build momentum, he and Ms. Murray introduced the legislation on Thursday along with two dozen leading Republican and Democratic co-sponsors — an almost unheard-of event on a major bill in this highly polarized environment.
Among Republicans backing the measure were Senators Charles E. Grassley of Iowa, long active in health policy; Lindsey Graham of South Carolina and Bill Cassidy of Louisiana, the authors of the most recent repeal plan; Bob Corker, Mr. Alexander’s Tennessee colleague who has been feuding with the president; and the group who thwarted Republican repeal efforts, John McCain of Arizona, Susan Collins of Maine and Lisa Murkowski of Alaska.
“I don’t believe Congress will want to fail to deal with a problem that will hurt millions of Americans if we allow it to continue,” Mr. Alexander said.
He and Ms. Murray said that they had built in protections to make sure the money went to consumers and that it was a mischaracterization to call it an industry bailout. They said both sides gave substantial ground in the talks. Ms. Murray told fellow Democrats this week that it was the toughest bargaining she had been involved in as a senator.
“It sends a powerful message that when members of Congress decide to get past our talking points and take a few steps out of our partisan corners, there is a lot we can agree on and a lot we can get done,” Ms. Murray said of the negotiation.
But to a large degree, the ultimate success of the talks depends on Mr. McConnell. He has been quiet so far about the bill and whether he would push it forward. The Senate would no doubt have to give it a strong vote for the plan to have any chance of moving through the House.
The support from a growing list of Republican co-sponsors is no guarantee that the Senate will consider the bill if Mr. McConnell views it as too politically risky. He refused to bring to the floor last year a bipartisan criminal justice overhaul that had broad support because of political considerations — despite the fact that Senator John Cornyn of Texas, the No. 2 Senate Republican, was a chief author.
Mr. Alexander has repeatedly predicted that some version of his new legislation will become law by the end of the year. For him to be proved right, he is going to need some help from his good friend Mitch McConnell.