CreditStephen Crowley/The New York Times
Mr. Upton predicted that the bill was “likely” to pass the House, a tremendous reversal of momentum for a measure that has twice been pulled back from a vote for lack of support.
Their announcement gave a big lift to Mr. Ryan and other Republican leaders as they tried to round up enough votes to push the bill through the House this week.
“We’ve got some momentum,” Mr. Ryan told a Wisconsin radio station on Wednesday morning.
Democrats and health care groups tried to slow that momentum. The liberal health advocacy group Families USA said another $8 billion would do little to improve the “high-risk pools” that could be set up by states to provide coverage to people with pre-existing medical conditions who could not find affordable insurance in the open market.
The American Medical Association and 10 organizations representing patients, including the American Heart Association and the advocacy arm of the American Cancer Society, reiterated their opposition to the House Republican bill on Wednesday, as did the retirees’ lobby AARP.
“None of the legislative tweaks under consideration changes the serious harm to patients and the health care delivery system” that the bill would cause, said Dr. Andrew W. Gurman, the president of the American Medical Association. The latest changes, he said, “tinker at the edges without remedying the fundamental failing of the bill — that millions of Americans will lose their health insurance as a direct result of this proposal.”
Senator Chuck Schumer of New York, the Democratic leader, also criticized the latest version of the legislation. “The proposed Upton amendment is like administering cough medicine to someone with stage-four cancer,” he said in a statement. “This Republican amendment leaves Americans with pre-existing conditions as vulnerable as they were before under this bill.”
If House Republicans can pass the bill, it would be a moment of redemption for both Mr. Ryan and Mr. Trump, who suffered a resounding political defeat in March when they failed to muster the votes to win approval of an earlier version.
The Affordable Care Act generally requires insurers to accept all applicants and prohibits them from charging higher premiums because of a person’s medical condition. Conservatives argue that this and other requirements of the 2010 health law drive up insurance costs.
At the insistence of conservative lawmakers, House Republican leaders agreed to let states apply for waivers allowing insurers to charge higher rates based on a person’s “health status.”
The original version of the Republican repeal bill would have established a $100 billion fund that states could use to help people pay for health care and insurance from 2018 to 2026. House leaders added $15 billion last month to help insurers pay claims for their sickest customers. Mr. Upton’s proposal would provide $8 billion over five years on top of that.
How far that $8 billion would go in providing coverage for people with pre-existing conditions is not clear. Mr. Upton’s proposal does not specify who would be eligible, how much of their costs would be covered or how much they would be expected to contribute in premiums.
How many states would seek waivers is difficult to predict.
But the fight over pre-existing conditions overshadowed a major reason the Congressional Budget Office estimated that the original bill would leave 24 million more Americans without health insurance after a decade: a rollback of the Affordable Care Act’s Medicaid expansion in states that adopted it. The House plans to vote for the latest version before the budget office can finish a fresh assessment of its cost and impact.
Representative Joe L. Barton, Republican of Texas, predicted his state “would lead the parade to opt out of all the federal mandates” in the Affordable Care Act.
But Representative Carlos Curbelo, Republican of Florida, said: “I would highly doubt that any governor, especially the governor of a large state like Florida, would seek a waiver. I just don’t think that any state would want to carry the burden of managing health care more than they already do, through Medicaid.”
Mr. Curbelo illustrated the fluid politics swirling around the repeal bill. In a Twitter post on Thursday morning, he said he had just told House Republican leaders that the bill “in its current form fails to sufficiently protect Americans with pre-existing conditions.” In a late afternoon interview, he said, “I do not yet have a position on the bill.” He wanted to hear more from Mr. Upton, a respected Republican voice on health care.
The Affordable Care Act set up a special health insurance program for people with cancer, heart disease and other serious illnesses, to provide coverage until 2014, when insurers were forbidden to discriminate against people based on their health status. Claims far exceeded Obama administration estimates, exhausting most of the $5 billion provided by Congress.
The average cost per enrollee was more than $32,000 a year in 2012, according to a federal report on the program, and the cost varied widely among states, from a low of $4,300 to a high of $171,900 per enrollee.
Mr. Upton and Sean Spicer, the White House press secretary, said they believed that the money in the bill would be adequate. “It’s our understanding that the $8 billion over the five years will more than cover those that might be impacted and, as a consequence, keeps our pledge for those that, in fact, would be otherwise denied because of pre-existing illnesses,” Mr. Upton said at the White House.
To qualify for assistance under the Upton proposal, a person would have to live in a state with an approved waiver, have a pre-existing condition and be uninsured because of a failure to maintain “continuous coverage.”
The House Democratic leader, Nancy Pelosi of California, said the money was a pittance compared with the likely need. “It’s a joke,” she said. “It’s a very sad, deadly joke.”
The latest amendments to the bill amount to “a hoax on pre-existing conditions,” Ms. Pelosi said. “If Republicans have their way, Americans with pre-existing conditions will be pushed off their insurance and segregated into high-risk pools where they face soaring cost, worse coverage and restricted care.”