Signs are seen on a barricade before a town hall meeting Wednesday with Rep. Leonard Lance (R-N.J.) in Branchburg, N.J. (Dominick Reuter/Reuters)

Congressional Republicans have been struggling for months to resolve one of the most vexing problems in their tortuous effort to replace the Affordable Care Act: What to do about the generous federal funding for states that broadened their Medicaid programs under the law, while not shortchanging the 19 states that balked at expansion?

Now, as the House begins to hone details of its legislative proposal, a possible compromise has emerged. It would temporarily keep federal dollars flowing to cover almost the entire cost of the roughly 11 million Americans who have gained Medicaid coverage but would block that enhanced funding for any new participants.

At the same time, the GOP approach would open a fresh spigot of aid for the states — all but one of which has a Republican governor — that eschewed the additional Medicaid money because of their elected officials’ antipathy to the law. This extra aid would probably go to hospitals with a large share of poor and uninsured patients.

The Solomon-like strategy is an attempt to calm fears of Republican governors in expansion states that abolishing the 2010 law would cost them hugely, while also satisfying new demands for equity from other GOP governors who opposed the expansion. ­Details of how the plan’s dual elements would be implemented, including their specific time frames and funding totals, are still coming together in the House Energy and Commerce Committee.

Within the context of the GOP’s broader repeal effort — and this week’s tumultuous town hall meetings around the country, at which lawmakers have been confronted by constituents scared of losing their health coverage — Republicans’ ideas for Medicaid’s future have drawn less public attention. Yet their proposals would significantly remake one of the nation’s largest entitlement programs, which serves more than 74 million lower-income Americans and accounts for half the additions to the insurance rolls that the ACA has brought about.

Amy Kuiken holds a sign during a health-care town hall meeting in Scranton, Pa., on Tuesday. Sen. Patrick J. Toomey (R-Pa.) was invited to speak but did not attend. (Butch Comegys/AP)

A similar approach is under consideration in the Senate. Sen. Rob Portman (R-Ohio), who is working toward a compromise to protect the roughly 700,000 Ohioans with Medicaid coverage ­under the ACA, said he is open to either a temporary extension or another way to subsidize their health insurance. “It’s necessary to figure out how to provide coverage, and that’s going to cost money,” he said Thursday.

The House committee also is moving forward with a plan to convert the rest of Medicaid to a system in which states would get a fixed sum of federal money for every resident who is enrolled. Such per capita funding, outlined by the chamber’s Republican leaders earlier this month, would be more restrictive than the way Medicaid has functioned since its birth as part of the Great Society legislation of the 1960s.

However, the allotments would be less rigid than block grants, which have been advocated for years by many conservatives and decried by liberals for their potential to reduce spending over time, prompting states to cut benefits or eligibility, or both. (Block grants might still emerge from Congress as an alternative that states could choose.)

This picture of the House’s ­behind-the-scenes work is based on interviews with several people familiar with the thinking of the Republican leadership. All spoke on the condition of anonymity since no plans have been announced.

The deliberations on Medicaid expansion are further along than other components of the law­makers’ thinking about how to shift government health policies in a more conservative direction, according to these individuals. The basic outline emerging from the Energy and Commerce Committee would “grandfather in” the adults currently on Medicaid-­ expansion rolls in 31 states and the District of Columbia. Still to be decided is whether the extra dollars for their coverage would last a specific length of time or continue as long as a beneficiary remained eligible.

As for non-expansion states, the extra money they would ­receive might come through an increase in “disproportionate share” payments the federal government has long given hospitals that treat a lot of poor patients. Or the government could increase its payments for Medicaid’s very poorest patients — a boost to Republican-led states across the South with large low-income populations.

The only Democratic-led state that has not expanded its Medi­caid program is Virginia, where Gov. Terry McAuliffe has been unable to overcome the resistance of the GOP-controlled legislature.

It’s unclear whether the plan would accomplish its goal of satisfying a range of the 35 GOP governors now in office, no matter the stance each has taken on expansion.

Gov. Scott Walker of Wisconsin, who chairs the Republican Governors Association, said at an event at The Washington Post on Friday that GOP governors are working closely with both lawmakers and the administration to determine how to transition those living above the poverty line off the expanded Medicaid rolls.

“Maybe I’m foolish, but I thought Medicaid is for people living in poverty,” Walker said.

But Gov. John Kasich (R-Ohio), who discussed health-care reform Friday with President Trump in the Oval Office, said in an interview that he considered the current compromise inadequate.

“I don’t think that paying hospitals for uncompensated care results with a healthier population,” he said, adding that expansion has provided resources for those struggling with addiction and mental illness. “Where are they supposed to go?”

The House Ways and Means Committee is doing parallel work on overall ACA replacement. But according to the several people familiar with the House leadership’s approach, a central idea under consideration there — new health-care tax credits — hit a snag this week when congressional budget analysts reported privately to the committee that the credits would cost the government a lot of money and would enable relatively few additional Americans to get insurance.

Those tax credits would replace subsidies the ACA provides people with incomes of up to 400 percent of the poverty level to help them afford health plans through marketplaces created under the law. The credits would be available to everyone who buys coverage on their own, wealthy or poor. But the Congressional Budget Office has concluded that the credits, as conceived at the moment, would be too small to help low-income people afford health plans. They also wouldn’t make much difference to affluent people, according to the CBO, since most of them already are insured.

Rep. Tom Cole (R-Okla.), an influential member of the Budget and Appropriations committees, said that while CBO models are not precise indicators of a bill’s actual fiscal impact, the new estimate should remind Republicans that they “should err on the side of being cautious, rather than make grandiose claims.”

Democrats fell into that trap when passing the ACA, he said. “We should not be overselling.”

When the ACA passed a polarized Congress in 2010, the idea was that about half the Americans who would gain health coverage would do so through the law’s insurance marketplaces. The other half would come through Medi­caid, which was to grow nationwide to include adults earning up to 138 percent of the poverty level. The federal government would pay the full cost of expanded coverage for the first three years, then a decreasing amount before settling at 90 percent by 2020.

In 2012, though, in a case brought by ACA critics who unsuccessfully challenged the law’s constitutionality, the Supreme Court ruled that each state had the latitude to choose whether to expand Medicaid. Nearly all ­Democratic-led states said yes; most with Republican governors opted out.

The latest polling by the Kaiser Family Foundation, released Friday morning, shows that Medi­caid expansion enjoys strong public support. Nationally, 84 percent of respondents — and 87 percent in the 16 expansion states with GOP governors — said it is important to preserve the greater federal funding provided by the ACA.