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.