Eager for a victory, the White House expressed confidence on Thursday that a breakthrough on the mired Republican health care bill could emerge in the House next week. The chamber’s GOP leaders, burned by a March debacle on the measure, were dubious and signs were scant that an emerging plan was gaining enough votes to succeed.

During a White House news conference, Mr. Trump said progress was being made on a “great plan” for overhauling the nation’s health care system, though he provided no details.

“We have a good chance of getting it soon,” Mr. Trump said. “I’d like to say next week.”

The White House optimism is driven largely by a deal brokered by leaders of the conservative Freedom Caucus and the moderate Tuesday Group aimed at giving states more flexibility to pull out of “Obamacare” provisions. A senior White House official acknowledged that it was unclear exactly how many votes Republicans had, but said House Speaker Paul Ryan, R-Wis., has told the White House that a vote could come together quickly.

Yet GOP lawmakers and aides to party leaders, conservatives and moderates alike were skeptical that the House would vote next week on the health legislation. They cited the higher priority of passing a spending bill within days to avert a government shutdown, uncertainty over details of the developing agreement and a need to sell the idea to lawmakers.

Mr. Trump said he planned to get “both” a health care deal and a spending bill.

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Many Republicans also expressed doubts that the health care compromise would win over enough lawmakers to put the bill over the top, especially moderates. The bill would repeal former U.S. President Barack Obama’s health care law and replace it with less generous subsidies and eased insurance requirements.

“Every time they move the scrimmage line, you risk losing other people who were ‘yes’ but this changes them to a ‘no,’” Rep. Dan Donovan, R-N.Y., said on Thursday of attempts to win over one end of the GOP spectrum without losing votes from the other side. The Staten Island centrist said he remained a no vote, partly because the legislation would increase Medicaid costs for New York City’s five boroughs.

The White House official and most lawmakers and GOP congressional aides who spoke were not authorized to discuss the internal process publicly and insisted on anonymity.

An outline of a deal has been crafted by Rep. Mark Meadows of North Carolina, who heads the hard-line Freedom Caucus, and New Jersey Rep. Tom MacArthur, a Tuesday Group leader. Vice President Mike Pence also played a role in shaping that plan, Republicans say.

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The plan would deliver a win to moderates by amending the GOP bill to restore Mr. Obama’s requirement that insurers cover specified services like maternity care. But in a bid for conservative support, states would be allowed to obtain federal waivers to abandon that obligation.

In addition, states could obtain waivers to an Obama prohibition against insurers charging sick people higher premiums than customers who are healthy. To get those waivers, states would need to have high-risk pools government-backed insurance for the most seriously ill people, a mechanism that critics say has often failed for lack of sufficient financing.

“It looks to me like we’re headed in the right direction,” Rep. Dave Brat, R-Va., a Freedom Caucus member, said on Thursday. He said that assuming the outline is translated into legislative text he backs and is added to the health care bill, he would now support the legislation and believes most of Freedom Caucus’ three dozen members would also back it.

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The Tuesday Group has roughly 50 members. But they don’t necessarily vote as a bloc, and it is unclear how many colleagues MacArthur would bring with him to such an agreement.

The White House is anxious to pass legislation quickly, partly because Mr. Trump will likely hit his 100th day in office without having signed a major piece of legislation. The White House is eager to cap the stretch with a significant achievement that could give the president a burst of momentum.

But House GOP leaders face the same problem that’s plagued them for seven years of trying to concoct a plan for repealing Obama’s 2010 law - The party’s conservatives and moderates are at odds over how to do it. With Democrats solidly opposed, Republicans can lose no more than 21 House votes to prevail, and House Speaker Paul Ryan short-circuited a planned vote last month because more than that would have defected.

That was a major embarrassment to Mr. Ryan and Mr. Trump, and House leaders are loath to bring a revised health care bill to the House floor unless they are convinced it would pass.

Mr. Ryan sent a mixed message about the bill’s prospects in remarks on Wednesday to reporters in London.

“It’s difficult to do. We’re very close,” he said, adding, “It’s just going to take us a little time.”