A Republican chairman on Thursday said talks to shore up Obamacare markets are “making good progress,” suggesting a renewed quest for bipartisan fixes for the struggling health law after GOP repeal efforts failed.

Senate Health Committee Chairman Lamar Alexander said he’s already trying to round up votes from fellow Republicans even as he negotiates final details of the package with Sen. Patty Murray, Washington Democrat.

It’s not clear what the appetite for shoring up Obamacare is among Republicans, including President Trump, who this week signaled another try at repeal early next year.

For his part, Mr. Alexander, Tennessee Republican, said Congress needs to put aside broader fights and pass a bill that helps the nearly 20 million Americans who buy insurance on their own. Many of them will face higher prices and dwindling choices in 2018, due to problems with Obamacare itself and Mr. Trump’s wavering commitment to it.

“This isn’t the ‘Patty and Lamar Show.’ This is a chance for us to see if we can find a significant amount of senators on both sides of the aisle who agree on a limited, bipartisan proposal to keep premiums down and avoid chaos in the individual market over the next two years, while we debate which direction we go with health care,” he said.

As envisioned, the compromise would fund cost-sharing payments that reimburse insurers who cover low-income Obamacare customers’ costs — a win for Democrats. Republicans would win more waivers for state governors to control their markets.

Senate Minority Leader Charles E. Schumer said he spoke to Mr. Alexander and Ms. Murray early Thursday, and ‘“they both informed me they are on the verge of an agreement.”

It’s a stark turnaround from nine days prior, when Mr. Alexander said the parties couldn’t find a path forward. Democrats cried foul at the time, saying Republicans left the bargaining table to clear the way for a repeal bill pushed by Republican Sens. Lindsey Graham of South Carolina and Bill Cassidy of Louisiana.

Lacking the votes, GOP leaders decided to yank the bill on Tuesday. Mr. Alexander and Mrs. Murray promptly resumed talks.

Prospects of an imminent deal enthused Democrats, who say constituents are being forced to pay higher premiums because insurers are hedging against the potential loss of cost-sharing payments. Mr. Trump is paying them now but hasn’t made a long-term commitment.

“Our healthcare system needs it, our constituents need it,” Mr. Schumer said. “They don’t want premiums to go up and coverage to go down, and it would be a great start for some bipartisanship in this place.”

The National Governors Association also applauded Mr. Alexander and Mrs. Murray, saying Congress must snap out of its partisan fights and “create a healthy future for our country.”

Still, it’s unclear if any stabilization plan can make it through the GOP-controlled Congress or win Mr. Trump’s signature.

Many conservatives say they have no interest in voting for insurer funds they view as a “bailout,” or propping up a law they think is hopelessly flawed. Conservatives pressure groups are holding their feet to the fire, as the GOP eyes another repeal push next year.

“Some lawmakers will reflexively demand a ‘bipartisan market stabilization’ in response to Obamacare’s deteriorating markets,” said Heritage Action CEO Michael Needham. “That is unacceptable, and Heritage Action will oppose efforts to prop up the failing law.”

Mr. Trump, meanwhile, has shown little interest in enforcing or promoting his predecessor’s signature domestic achievement.

Beyond threats to cut off cost-sharing payments, he’s made it harder for the IRS to enforce Obamacare’s “individual mandate” requiring people to hold health insurance and slashed outreach funding ahead of open enrollment this fall.

Democrats say the uncertainty is causing real harm and will burn Republicans who’ve professed an affinity for market-oriented health care.

“What does the private sector care about the most? Certainty and predictability,” said Sen. Ron Wyden, Oregon Democrat. “For months and months, the president of the United States has been pouring gasoline on the fires of uncertainty in the private health care marketplace.”

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