He and Senator Patty Murray of Washington, the top Democrat on the panel who has been a productive negotiating partner with Mr. Alexander in the past, have agreed to convene hearings when the Senate returns in September and to try to push some minimalist legislation through Congress by the end of the month.

Even a small bill would be a feat in a Congress that has delivered so few results. But Mr. Alexander, a durable believer in the legislative process, sees it as a possibility, with the alternative being a failure that is certain to rock already reeling individual insurance markets.

“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.