Senate Republicans, largely those from states that chose to expand their Medicaid programs under the Affordable Care Act, so far have not seemed susceptible to pressure from leaders and Mr. Trump, listening instead to governors and constituents concerned about significant reductions in benefits.

Part of the bill’s problem is time itself. Much has changed in the years since the Affordable Care Act passed, with millions of Americans, many in red states, now getting health insurance as a result of the law, as well as treatment for the prescription drug addictions that have plagued scores of communities.

“My goal for this whole process was to help the people the law harmed and not harm the people it helps,” said Representative Dan Donovan, Republican of New York. At the same time, a fair number of conservatives would like to see those benefits greatly reduced, the central tension of the Republican debate.

As a result, it remains difficult to imagine a bill that could find its way out of the Capitol to Mr. Trump’s desk, given the broad disparities in what Republicans now seek.

Even if they can come together, House Republicans risk making the same mistakes Democrats made in the beginning of Mr. Obama’s term, when they pushed through what came to be known as Obamacare. That achievement, monumental at the time, ended up dragging down a once formidable Democratic majority and reducing the ability of Democrats to pass more legislation during his presidency.

Yet if the bill fails, Republicans in the House could end up like House Democrats under President Bill Clinton, who passed a controversial energy tax that was reduced to rubble in the Senate, but remained an albatross for Democrats in the 1994 elections.