The Republicans’ repeal bill, according to the Congressional Budget Office, would have left an additional 24 million Americans without insurance by 2026, a major worry for moderate Republicans. It would also have left in place regulations on the health insurance industry that are anathema to conservatives.

Mr. Ryan declined to say what might be in the next version of the Republicans’ repeal bill, nor would he sketch any schedule for action. But he said Congress needed to act because insurers were developing premiums and benefit packages for health plans to offer in 2018, with review by federal and state officials beginning soon.

The new talks, which quietly began this week, involve Stephen K. Bannon, the president’s chief strategist, and members of two Republican factions that helped sink the bill last week, the hard-right Freedom Caucus and the more centrist Tuesday Group.

Any deal would have to overcome significant differences about how to rework a law that affects about one-fifth of the American economy. Those differences were so sharp that they led Mr. Trump and Mr. Ryan to pull the bill just before the House was to vote on it.

Still, Republican members of Congress said they were hopeful.

“I think everyone wants to get to yes and support President Trump,” said Representative Dave Brat of Virginia, a Freedom Caucus member. “There is a package in there that is a win-win.”

Representative Raúl R. Labrador of Idaho, also a Freedom Caucus member, said he hoped the discussions would yield a compromise after a divisive debate that revealed deep fissures within the party. “I think we will have a better, stronger product that will unify the conference,” Mr. Labrador said.