The Koch-backed groups, in a statement issued late Wednesday, said they were committing seven figures to the effort.
“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 at the White House with top administration officials, including Mr. Pence and Kellyanne Conway, a senior adviser to Mr. Trump.
The lawmakers said the bill, drafted by House Republican leaders, did not do enough to lower health insurance costs by reducing federal insurance regulations, including standards that define minimum benefits.
“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. The conservatives were upset over the failure of the House bill to repeal a set of regulations in President Barack 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, who is 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, and I feel confident that when the vote comes up, we’ll have the votes,” he said Wednesday, adding: “There is no Plan B. There’s Plan A and Plan A. We’re going to get this done.”
Supporters of the legislation are counting on Mr. Trump to exert influence on reluctant conservatives, betting that such holdouts will ultimately shy away from the prospect of public confrontation with the president.
Representative Pete Sessions, Republican of Texas and chairman of the Rules Committee, presided over a daylong session of the panel to set the rules for debate on the House floor.
“These Freedom Caucus guys are awesome,” said Mr. Sessions, a trusted lieutenant of House leaders. “They have been consistent in their language, in their behavior, in their requests and in their discussions.”
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