WASHINGTON — As they explore possible fiscal deals, President Obama and Congressional Republicans have quietly raised the idea of broad systemic changes to Medicare that could produce significant savings and end the polarizing debate over Republican plans to privatize the insurance program for older Americans.

While the two remain far apart on the central issue of new tax revenue, recent statements from both sides show possible common ground on curbing the costs of Medicare, suggesting some lingering chance, however small, for a budget bargain.

Mr. Obama assured House and Senate Republicans during recent separate visits that he could support specific cost-saving changes to Medicare and deliver Democratic votes, though only as part of a “balanced” package that had additional revenues.

Several changes are likely to once again be in his annual budget, which will be released on April 10, after Congress returns from its break. Mr. Obama also plans a dinner with Senate Republicans that night.

In particular, participants say, the president told House Republicans that he was open to combining Medicare’s coverage for hospitals and doctor services. That would create a single deductible that could increase out-of-pocket costs for many future beneficiaries, but also could pay for a cap on their total expenses and reduce the need to buy Medigap supplementary insurance.

Representative Eric Cantor of Virginia, the No. 2 House Republican, proposed much the same in a speech in February. “We should begin by ending the arbitrary division between Part A, the hospital program, and Part B, the doctor services,” he said. “We can create reasonable and predictable levels of out-of-pocket expenses without forcing seniors to rely on Medigap plans.”

While Mr. Cantor’s proposal got little attention at the time, its echo by Mr. Obama hints at a new route toward compromise — in contrast with the budget that House Republicans passed this month that has no chance of Senate approval.

At a time when retiring baby boomers and mounting medical prices have made federal health care spending the biggest single driver of the nation’s rising debt, the House budget from Representative Paul D. Ryan, Republican of Wisconsin, would transform Medicare into a voucherlike system known as premium support, which Mr. Obama and Democrats adamantly oppose. But Mr. Cantor, like Mr. Obama, is suggesting cost-saving changes within the existing Medicare program.

To Senator Mark R. Warner of Virginia, a Democrat who has long led a bipartisan group of senators seeking a fiscal deal, such a proposal is the sort of newer idea needed for the parties to stop the “stale arguments” that after three years have turned their budget battling into “World War I trench warfare.”

“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.”

Both the administration’s and Mr. Cantor’s interest in restructuring Medicare’s Parts A and B dates to 2011, when various proposals were considered by a deficit-reduction group headed by Vice President Joseph R. Biden Jr. that included Mr. Cantor.

The goal is to discourage people from seeking unneeded treatments, shrink health spending and offset the costs of a cap on beneficiaries’ total out-of-pocket costs. Such a cap would reduce beneficiaries’ need for extra insurance. About 90 percent of beneficiaries in the traditional Medicare program have supplemental coverage through Medigap policies, employers’ retiree plans or Medicaid for low-income people.

Many health-policy economists have called for creating a single, unified deductible. The current two deductibles reflect separate legislative tracks that came together in the creation of Medicare in 1965. The deductible for Part A hospital care is relatively high ($1,184 this year), while that for Part B doctor care is relatively low ($147). Patients also have co-payments for many services.

Despite the bipartisan interest, the politics of merging Part A and Part B are complicated.

Glenn M. Hackbarth, chairman of the Medicare Payment Advisory Commission, a group of nonpartisan experts that advises Congress, said a combined deductible could increase costs for those who use only doctor and outpatient services — a majority of beneficiaries in any year. It could reduce costs, he said, for the roughly 20 percent who require hospitalization.

Proponents, including some in the administration, acknowledge the political risks of increasing most beneficiaries’ costs, even in exchange for capping their total costs, as in cases of catastrophic illness. A 1988 law protecting against catastrophic costs caused such an outcry among older Americans, who faced an extra tax, that Congress quickly repealed it.

But administration officials say the 1988 law affected current beneficiaries, while Mr. Obama would apply any changes only to people becoming eligible for Medicare after 2016.

So far, the changes the president has proposed do not go as far as a single deductible and a cap on catastrophic costs. Instead, Mr. Obama has called for increasing the Part B deductible, which has risen much less than medical costs. He also proposed that beneficiaries pay something for home health care, which is among Medicare’s fastest-growing and most fraud-prone expenses; people just released from the hospital would be exempted.

Third, Mr. Obama proposed a 15 percent surcharge on Medigap plans that cover all or nearly all of a beneficiary’s initial annual expenses. Economists say that such coverage leaves beneficiaries insensitive to costs, increasing Medicare’s spending and the premiums beneficiaries pay.

###