The bill would also boost funding for Obamacare enrollment outreach efforts, and require federal health officials to draft rules that would allow health plans to be sold across state lines.
The deal was put together in recent days after the Trump administration last week canceled cost-sharing reduction payments to Obamacare insurers that have been made since 2014.
Those payments reimburse insurers for discounts to health costs for low-income customers that the Affordable Care Act requires they grant, by law.
The Trump administration cited a legal opinion from Attorney General Jeff Session, who said the payments could not be made without an appropriation by Congress. The Republican-led Congress for years has refused to make such an appropriation.
Insurance prices for individual plans in coming years are projected to jump higher than they otherwise would have because of Trump's decision.
And the move will add almost $200 billion to the federal budget deficit, because other Obamacare subsidies will increase to offset the effect of higher premium prices.
On Wednesday, nine governors, four of them Republicans, sent congressional leaders a letter asking that they "quickly pass legislation to stabilize our private health insurance markets."
The governors noted that the elimination of the CSR payments will cause "significant financial losses" for insurers, "which could force them to withdraw from the marketplace, or, in some states, request significant rate increases."
Also Wednesday, a coalition of more than dozen states' attorneys general filed a court action seeking to force the federal government to continue paying the insurers. The coalition said they payments are mandated by the Affordable Care Act.
An Economist/YouGov poll found that 53 percent of respondents disapproved of Trump's decision to end the reimbursements, with just 31 percent in favor.
A total of 52 percent of respondents to the survey believed the payments should be restored by Congress. Only 21 percent said Congress should not do so.
In addition to Alexander, McCain, Cassidy and Graham, the bill's Republican co-sponsors are Mike Rounds of South Dakota, Susan Collins of Maine, Joni Ernst of Iowa, Lisa Murkowski of Alaska, Charles Grassley of Iowa, Johnny Isakson of Georgia, Richard Burr of North Carolina and Bob Corker of Tennessee.
The Democrats co-sponsoring the bill are Jeanne Shaheen of New Hampshire, Joe Donnelly of Indiana, Amy Klobuchar and Al Franken of Minnesota, Heidi Heitkamp of North Dakota, Joe Manchin of West Virginia, Tom Carper of Delaware, Tammy Baldwin of Wisconsin, Claire McCaskill of Missouri and Maggie Hassan of New Hampshire.
Angus King of Maine, an independent senator who caucuses with the Democrats, is also sponsoring the bill.
Brad Woodhouse, director of the Obamacare advocacy group Protect Our Care Campaign, said, "The Alexander-Murray bipartisan health-care legislation gained significant momentum with today's announcement that this bill has strong bipartisanship support — vastly more than the partisan Senate repeal efforts that never mustered even 50 votes."
"It's time for all Republicans to get behind this bipartisan approach that stabilizes the market," Woodhouse said.