“It has to be simple if we are to get bipartisan agreement by mid-September on an issue that has divided the parties so much,” he said. Stabilizing the markets for a year, he said, would provide breathing room to “tackle bigger issues” on health care.

Mr. Alexander’s political and policy challenges are formidable. First, he would have to get a consensus on his own committee, which ranges from Senator Rand Paul, Republican of Kentucky, on the right, to Senator Bernie Sanders, independent of Vermont, on the left, with a bit of everything else in between. Then he would have to get it through the full Senate, where nothing to do with health care has been able to attract a majority. Then the measure would go to the House, where resistance is even stronger to anything that resembles an effort to prop up the health care law.

Finally, there is the White House, where President Trump has suggested he might unilaterally terminate the funding Mr. Alexander wants to preserve and let the current health program collapse — an idea the senator thinks is a bad one. His opposition could put him in the line of fire from a president he barely knows.

But Mr. Alexander is the rare senator who has strong relationships with Senator Mitch McConnell, the Kentucky Republican and majority leader, as well as Senator Chuck Schumer of New York, the Democratic leader. Those ties could prove useful. But he is already under attack from conservatives framing his effort as a bailout for insurers — a critique meant to resonate with voters still angry about the 2008 bank bailout.

“The Senate’s inability to produce 51 votes for a piece of legislation that delivers on a seven-year campaign promise to repeal and replace Obamacare is not license for a bipartisan bailout of a failing law,” Michael Needham, head of Heritage Action, a conservative advocacy group affiliated with the Heritage Foundation, said. “Obamacare is becoming a zombie law, and throwing more taxpayer money at Zombiecare is unacceptable.”

Mr. Alexander, who has run for president twice, served as education secretary and was a two-term governor and a university president before joining the Senate in 2003, realizes he is going to come under fire for doing anything seen as sustaining the health care law. He is undeterred.