States could request an exemption from the rule intended to ensure that people with pre-existing conditions could not be charged prohibitive premiums — but only if those states establish a high-risk insurance pool.

“The plan gets better and better and better, and it’s gotten really, really good, and a lot of people are liking it a lot,” Mr. Trump said. Asked if a health bill could pass as Congress tries to avert a government shutdown, the president said, “I think we’ll get both.”

The complications that remain in the bill are likely to be far too difficult to finesse at the same time the House and Senate press to pass a giant spending bill. Tussles over the spending deadline — including possible debates over top administration priorities like a border wall and money for immigration enforcement officers — are expected to consume the Capitol.

And Democrats — whose votes will be needed to keep the government open — will have their own demands, most importantly billions of dollars to lower out-of-pocket spending for low-income Americans purchasing health coverage on the Affordable Care Act’s online marketplaces.

Senior Republicans appear unconvinced that a revised health care bill would ensure passage in the House. Mr. Donovan, an opponent of the original Republican health care bill, said the proposed amendment “really doesn’t address the concerns that I had.”

Representative Charlie Dent, Republican of Pennsylvania and a leader of the moderate House Tuesday Group, said it “does nothing to change my views.” He lamented any focus “on an arbitrary 100-day deadline.”

The changes — proposed by Representative Tom MacArthur, Republican of New Jersey and co-chairman of the Tuesday Group — come as Republicans face anger from supporters over their failure to act on longstanding campaign pledges, as well as from defenders of the Affordable Care Act.