The starting point for the current negotiations is President Obama’s most recent budget request, which proposed legislation that would save $300 billion, or 4 percent of projected Medicare spending, over 10 years.

By contrast, Republicans in Congress are seeking savings of $400 billion to $600 billion, at least some of which should come from beneficiaries, they say.

Members of the Medicare Payment Advisory Commission, an influential panel that advises Congress, see many opportunities to rein in costs, and they say that financial pressure on providers could make them more efficient without harming the quality of care. At a meeting of the panel earlier this month, one commission member, Scott Armstrong, president of Group Health Cooperative, a nonprofit health system in Seattle, said Medicare spent “too much” on inpatient hospital care — $117 billion last year. “In an efficient system,” he said, “we wouldn’t be spending that kind of money on hospital services.”

Although Congress may leave the details of Medicare savings to be worked out next year, there is already discussion of cutting special payments to teaching hospitals and small rural hospitals. Lawmakers are also considering reducing payments to hospitals for certain outpatient services that can be performed at lower cost in doctors’ offices. Medicare pays substantially higher rates for the same services when they are provided in a hospital outpatient department rather than a doctor’s office. The differential added $1.5 billion to Medicare costs last year, and as hospitals buy physician practices around the country, the costs are likely to grow, the Medicare commission says.

The savings contemplated by Mr. Obama and Mr. Boehner are substantially larger than the Medicare savings that would be produced by automatic across-the-board cutbacks scheduled to start next month if Congress does not intervene. Those Medicare savings have been estimated at $123 billion from 2013 to 2021. Some hospital executives favor the automatic cuts as more equitable — and less painful — than some of the specific reductions being contemplated.

Hospital administrators and others warn of potential hospital closings, shutting down of unprofitable services like hospitalization for psychiatric care and less access to medical care for the most vulnerable if the cuts are too deep. Nancy M. Schlichting, the chief executive of the Henry Ford Health System in Detroit, says severe cuts might make it harder for hospitals like hers to treat patients without insurance. “It’s a big question whether we can continue to do that,” she said. “We would have to make tough decisions.”