Alexander, aides said, has an ambitious goal of moving quickly from hearings to drafting legislation that would, at minimum, guarantee the continued payment of cost-sharing reduction subsidies to insurance companies and allow states more flexibility to adjust insurance rules through an existing provision of the Affordable Care Act known as Section 1332. “I’m looking for the simplest bill possible that Republicans and Democrats can agree on that will stabilize the individual insurance market,” Alexander told reporters earlier this month.

The cost-sharing reduction payments are a top priority for Democrats since they are a part of Obamacare designed to help insurers afford the cost of insuring expensive patients without jacking up premiums for everyone else. Because the payments have been tied up in litigation brought by House Republicans against the Obama administration, President Trump has threatened to withhold them on nearly a monthly basis, only to make them at the last minute. Insurers have cited the uncertainty in announcing their decisions to leave certain markets and raise premiums, handing Democrats ammunition to accuse the Trump administration of sabotaging the law. Even most Republicans have encouraged the president to make the payments, but new legislation is the only way to guarantee them since the legal question at issue is whether Congress authorized the money in the first place.

Granting states more flexibility from Obamacare regulations is a must-have for Republicans, and Democrats have signaled they are willing to at least listen to ideas as long as the changes do not threaten coverage for people with preexisting conditions or essential health benefits under the law. Murray’s standard, according to an aide, is that Congress “cannot go backward” by removing core protections currently in place. While both parties have many other ideas about improving Obamacare, any initial bipartisan proposal is likely to be narrow. “We’re very much looking for a smaller bill with lots of agreement,” a Republican aide told me.

One key reason is time: Alexander wants to act by September 27, the deadline for insurers participating in the Obamacare exchanges to finalize their plans. Congress also has a number of items it must pass by the end of the month, and the best chance for bipartisan health-care legislation to be enacted might come if it is attached to one of those bills. The Obamacare marketplaces have already seen some improvement in recent weeks as insurers have stepped in to fill all of the counties that previously lacked coverage, undermining the GOP argument that the law is “collapsing.” But legislative action in the next month could limit premium increases that companies are considering if congressional subsidies are not guaranteed.

Alexander and Murray face plenty of hurdles next month, the biggest of which is that many Republicans are simply not ready to move on from repeal. Trump is still steaming at the Senate’s failure, lobbing insults at Majority Leader Mitch McConnell and urging Republicans to try again and again. Senator Lindsey Graham of South Carolina is pitching the White House and fellow senators on a proposal he’s written with Senators Bill Cassidy of Louisiana and Dean Heller of Nevada that would repeal Obamacare’s individual and employer insurance mandates while devolving the rest of the law to the states. It keeps all of the ACA’s taxes and redirects a chunk of money to states that chose not to expand Medicaid. “I think we’re going to pull a rabbit out of a hat on health care reform,” Graham told the conservative commentator Hugh Hewitt on Thursday.