“You’ve got a whole lot of folks on the Republican side saying, ‘Well, we don’t really like what Ryan has done — premium support — but we want systemic reform,’ ” Mr. Warner said at a round table hosted by Bloomberg News.

Mr. Obama’s openness to Medicare changes seemed to be news to many Republicans, even though he first proposed detailed ideas in 2011. Republicans often accuse the president of opposing changes in entitlement spending while focusing on raising taxes, an attack that ignores his proposals but also reflects how little Mr. Obama has talked about them.

Still, the same hurdle to compromise stands: The president and his party will not support even his Medicare proposals unless Republicans agree to raise taxes on the wealthy and some corporations. Without that trade-off, common ground on Medicare will remain unplowed.

“The president has said this to the Republicans: ‘You want to do entitlement reform? I do, too. I can produce entitlement reform and bring Democrats to the table, because I am a Democratic president. And so I’m ready to sit down with you and work out an approach,’ ” Senator Richard J. Durbin of Illinois, a Democratic leader, said at a recent forum hosted by The Wall Street Journal.

Many Republicans remain distrustful of Mr. Obama. Yet when they speak of altering Medicare, not replacing it, it is clear that they share some concerns about the existing program.

Representative Kevin Brady, Republican of Texas and chairman of a health subcommittee, said the structure of the traditional fee-for-service Medicare is “outdated and confusing.”

“Can you imagine a world in which someone has to buy hospital and nursing home coverage from one insurance company, physician office coverage from another insurance company, prescription drug coverage from yet another company, and likely supplemental coverage from a fourth insurance company?” Mr. Brady asked. “This is exactly how the current Medicare benefit is designed.”