“We’re encouraged tonight, just based on the real willingness of not only the White House, but our leadership, to make this bill better,” Mr. Meadows said, crediting the personal involvement of Mr. Trump and Mr. Pence.
But Representative Charlie Dent, Republican of Pennsylvania and a leader of a moderate bloc of lawmakers known as the Tuesday Group, said late Wednesday night that he would oppose the bill.
“I believe this bill, in its current form, will lead to the loss of coverage and make insurance unaffordable for too many Americans, particularly for low-to-moderate income and older individuals,” Mr. Dent said.
And the powerful conservative network funded by the billionaire brothers Charles and David Koch issued a direct challenge to the president and said that it would provide financial support to members who voted against the plan.
“We will stand with lawmakers who keep their promise and oppose this legislation,” said James Davis, executive vice president of Freedom Partners, the umbrella organization responsible for the Koch brothers’ political efforts.
About two dozen conservative Republicans, including Freedom Caucus members, met Wednesday with top administration officials, including Mr. Pence and Kellyanne Conway, a senior adviser to Mr. Trump.
“I don’t think they changed any minds,” Representative Randy Weber, Republican of Texas, said after the meeting.
The tenacity and persistence of the conservatives appeared to give them outsize influence as Mr. Ryan struggled to round up votes for the repeal bill, which faces solid opposition from House Democrats. Supporters of their bill have put their faith in Mr. Trump, whose young presidency could be badly damaged by a public and consequential loss.
“When the president calls someone and says, ‘I need your vote on this,’ it’s very hard to say no to the president of the United States when this torpedoes our entire conference, Trump’s entire presidency, and we end up losing the Senate next year and we lose members in the House,” said Representative Chris Collins, Republican of New York and a top Trump supporter in the House.
But conservative opposition was over substance, not politics. Conservatives are upset over the failure of the House bill to repeal a set of regulations in Mr. Obama’s signature health law, which require insurers to cover a base set of benefits, like maternity care, preventive services, wellness checkups and rehabilitative services. These “essential health benefits” raise the cost of insurance and prevent companies from offering stripped-down options, the conservatives say.
“How can you talk about repealing the Affordable Care Act, Obamacare, without repealing the essential health benefits?” asked Representative Scott Perry, a Pennsylvania Republican who attended the meeting with Mr. Pence.
Republican leaders say that if the House makes such changes to the bill, it could imperil their ability to push the legislation through the Senate using expedited procedures that neutralize the threat of a filibuster.
Representative Mike Simpson, Republican of Idaho, likened the swirling cloud of uncertainty to the situation in November 2003, when the House approved a bill adding prescription drug benefits to Medicare after a roll-call vote that lasted nearly three hours in the middle of the night. The bill passed, 220 to 215, after House Republican leaders put down a conservative rebellion.
“It’s tough to pass controversial things, especially when Republicans have different ideas,” Mr. Simpson said. Eventually, he predicted, House leaders will get the votes they need, though they may need to tweak the repeal bill.
Representative Scott DesJarlais, Republican of Tennessee, said the administration tried to sell the House bill, known as the American Health Care Act, by arguing that it could be improved later in the Senate. But House members rarely relish handing their political fate to the other chamber.
“I am more skeptical,” Mr. DesJarlais said. “I like to see what I’m going to get when I vote for it, not promises that I get later.”
Asked if supporters of the bill had the votes to pass it in the House, Mr. DesJarlais said, “I don’t think they do.”
A spokeswoman for the Freedom Caucus, Alyssa Farah, said that more than 25 members of the caucus were “no” votes on the health care measure — enough to sink the bill in the House, though that count could not be independently verified.
Representative Andy Harris, Republican of Maryland, said that despite recent changes to the health care bill, he was unable to vote for it.
“This legislation simply won’t lower premiums as much as the American people need, and lowering the cost of coverage is my primary goal,” said Mr. Harris, an anesthesiologist and member of the Freedom Caucus.
House leaders were also contending with opposition from more moderate Republicans worried about the toll that the health bill could take in their districts. Representative Dan Donovan of New York, who attended a meeting at the White House with Mr. Trump on Tuesday, said Wednesday that he would vote against the bill.
“Recognizing that the status quo is failing isn’t, on its own, a compelling reason to vote ‘yes’ on the current replacement plan,” said Mr. Donovan, the only Republican House member from New York City.
Sean Spicer, the White House press secretary, said he was sure the House would pass the repeal bill. “Slowly but surely we’re getting there,” he said. “There is no Plan B. There’s Plan A and Plan A. We’re going to get this done.”