WASHINGTON — Two senators leading bipartisan talks on health insurance are hoping to have a proposal ready by the end of next week that would stabilize Obamacare’s individual insurance markets for the next two years.

Sen. Lamar Alexander, R-Tenn., said Friday that he and Sen. Patty Murray, D-Wash., are looking to have legislation they can agree upon ready by Monday and then finish their talks with other senators from both parties next week.

By the end of next week, they want to have the proposal in the hands of Senate Majority Leader Mitch McConnell, R-Ky., and Senate Minority Leader Chuck Schumer, D-N.Y.

“What we’re trying to do is not just see whether Sen. Murray and I can agree, but whether the two of us can find a significant number of Democrats and Republicans who can agree on a limited, bipartisan proposal that could actually pass,” said Alexander, chairman of the Senate Health, Education, Labor and Pensions Committee.

Murray, the committee’s top Democrat, appeared optimistic about the discussions.

“After all the partisanship we’ve seen from Republicans on health care, I’m glad we’ve been able to restart our conversations about ways to actually make health care work better for families — beginning with steps to help lower premiums — and I’m hopeful we can reach a final agreement soon,” she said.

While negotiations are still ongoing, Alexander and Murray are looking to give states more flexibility in the type of policies they can approve and to extend for two years the federal cost-sharing payments that enable insurance companies to reduce premiums for lower- and middle-class Americans. President Trump has threatened to stop the payments, which are worth about $7 billion this year.

Extending the federal cost-sharing received almost universal support from state insurance regulators, governors, health care executives and others who testified earlier this month at four hearings before the committee.

Insurance rates for 2018 were finalized on Wednesday, but Alexander said he thinks there’s still time to provide relief for consumers who will face higher premiums next year.

“Insurance companies would have to take extraordinary action to either rebate premiums or give consumers credit,” he said. “But I’m not going to favor paying the cost-sharing payments to insurance companies for 2018 unless consumers get the benefit instead of the insurance companies.”

By 2019, insurance rates could begin to go down in most states based upon the combination of proposals that are under consideration, Alexander said.

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Alexander and Murray began working toward a bipartisan deal after the collapse in July of the GOP’s efforts to repeal the Affordable Care Act.

“To put it bluntly, our goal is to avoid chaos in the individual market, where premiums are skyrocketing and up to 16 million people literally might not be able to buy insurance at all in that market because no company was willing to sell insurance,” Alexander said.

He briefly pulled the plug on the talks last week after Senate GOP leaders began pushing a separate bill by Republican Sens. Lindsey Graham of South Carolina and Bill Cassidy of Louisiana to restructure Obamacare.

Alexander reopened the discussions with Murray when the Graham-Cassidy bill collapsed because it didn’t have enough votes to pass.

Even if the senators reach a bipartisan deal, it is uncertain whether such an agreement could win enough votes in the Senate and the House to become law. The measure would most likely require 60 votes to pass in the Senate.

Sen. Roy Blunt of Missouri, a member of GOP leadership, said he believes the Senate could pass a bipartisan health care fix but wasn’t sure whether it could clear the House or get President Trump’s signature.

“I believe they’ll be able to come up with a proposal, and a number of senators will sponsor it,” Blunt said of the negotiations between Alexander and Murray. “I don’t have a sense of where the president’s going to wind up or where the House is. I think the Senate would likely be able to get something done there. But whether the other two essential players are willing to go in that direction, I don’t know.”

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In the House, Rep. Mark Walker, R-N.C., who chairs the conservative, 150-plus-member Republican Study Committee, signaled an openness to the talks going on in the Senate. “The House will be receptive to anything that allows specifically the middle class a little bit of relief from the premium hike,” he said.

But Rep. Mark Meadows, R-N.C., who chairs the hardline House Freedom Caucus and has been a key negotiator in trying to bring health care legislation further to the right, had his doubts that Alexander and Murray would be able to produce legislation that could make it past the House.

“It depends on what it is,” Meadows said. “Obviously, we’re open to any bipartisan solution, and yet probably the biggest thing that I see, with most of what they’re talking about, is just extending (cost-sharing) payments ... without a transition to something that lowers premiums.”

That, Meadows said, “is pretty much a nonstarter” in the House.

Alexander said he has spoken to House Speaker Paul Ryan, R-Wis., as well as Senate leaders McConnell and Schumer, about the need to pass a short-term fix for the individual market.

“My argument to them is … that we should take whatever steps we could to avoid the chaos that would occur if we don’t act,” he said. “Whether they’re for or against Graham-Cassidy, I think Republicans and Democrats are likely to agree with that.”

Contributing: Deirdre Shesgreen and Eliza Collins