Sens. Lamar Alexander and Patty Murray, the bipartisan duo behind a bill to stabilize Obamacare, on Thursday formally introduced their legislation, which included 24 co-sponsors from both parties.

Despite receiving inconsistent support from the White House and being panned by House leaders, Alexander, a Tennessee Republican, said he was certain the bill "would become law in some fashion before the end of the year." If the bill isn't passed separately, some lawmakers have said, it could be part of a spending measure in which Democrats have more leverage.

The co-sponsors of the legislation are split evenly between Democrats and Republicans, and the bill would need 60 votes in the Senate for passage. Some of the 12 GOP co-sponsors include Sens. Susan Collins of Maine, Lisa Murkowski of Alaska and John McCain of Arizona, who had opposed Obamacare repeal efforts in the summer, dooming efforts to overhaul the law.

Other GOP co-sponsors included Sens. Lindsey Graham of South Carolina and Bill Cassidy of Louisiana, who backed a last-ditch effort at Obamacare repeal that would have overhauled the law by transferring its revenue to states in the form of block grants so they could set up their own healthcare systems.

Conservatives have said they view the Obamacare package as a "bailout" to health insurance companies and say that it does not fulfill Republicans' seven-year promise to repeal and replace the law. The White House has spelled out President Trump's suggestions for the bill, including that it provide "relief" from the individual and employer mandates and that it allow more people to band together to form "association health plans," which would be similar to an executive order he signed a week ago.

But Alexander pressed the need for the payments and for a deal in remarks on the Senate floor. He noted Trump called him twice to encourage bipartisan efforts.

Alexander said the president told him, "I think I can get block grants to replace Obamacare [referring to Graham-Cassidy], but I don't want people to suffer in the meantime." According to Alexander, he said the president encouraged him to negotiate with Murray and urged him to use cost-sharing subsidies, which Trump ended last week.

Murray, D-Wash., insisted during her floor remarks that they would continue to push for passage.

Alexander and Murray's bill would fund cost-sharing reduction payments for insurers for the rest of the year and for two additional years, and would allow more people to buy "copper" plans with lower premiums and higher deductibles. It also would accelerate states' ability to obtain waivers for how they implement Obamacare.

Senate Minority Leader Chuck Schumer has praised the Alexander-Murray bill. He initially said that Democrats were not open to revisiting negotiations over its details, but on Thursday morning on the Senate floor said that changes could be considered to specify that the cost-sharing funds offer relief to Obamacare customers and not insurers.

Insurers are still required to pay for cost-sharing funds under Obamacare and will increase rates on premiums to make up for the difference. Low-income people with subsidies receive assistance to pay for those increases, but middle-income people who have Obamacare plans, whether purchased on or off the exchanges, will face higher costs.

Alexander pushed back against the "bailout" label about the cost-sharing funds during his floor remarks, saying they were meant to help low-income people. The bill requires each state to detail how they would ensure that insurance companies aren't double dipping by both increasing premiums and accepting the cost-sharing funds.

Still, he appeared open to making changes.

"We are certain it can be improved," he said. "We are looking forward to working with those who want to improve it."

Editor's note: This story has been updated to reflect that Sen. Chuck Schumer is the Senate minority, not majority, leader.