Medicare and Medicaid insure more than 100 million people, account for 23 percent of all federal spending and are likely to be an important part of any budget deal. Military spending, which accounts for about 20 percent of federal expenditures, is likely to be included as well.

Most Republicans have ruled out tax rate increases to reduce the deficit. Mr. Obama has rejected the idea of Medicare vouchers, Medicaid block grants or any rollback of the new health care law. But he and the Republicans say they still hope to find some common ground.

Mr. Obama has embraced the goal of reducing deficits by a total of $4 trillion over 12 years — an ambitious goal that suggests the size of any grand bargain.

In a speech in April, Mr. Obama offered to slow the growth of Medicare and Medicaid without cutting benefits. He said his ideas would save $340 billion over 10 years and a total of nearly $500 billion in the two programs by 2023. His numbers quickly became a starting point in the negotiations.

As for Medicaid, administration officials have indicated that they could accept savings of $100 billion or more over 10 years, much to the dismay of many House Democrats. The lawmakers say the cuts would impair access to care for the poor and shift costs to the states, which are facing a huge expansion in Medicaid eligibility and enrollment, scheduled to start in 2014 under the new health care law.

While insisting on new revenue at his news conference last week, Mr. Obama also said, “We’ll have to tackle entitlements,” adding that “health care cuts” need to be part of any deal.

Senator Joseph I. Lieberman, the Connecticut independent, described a fiscal and political imperative: “We can’t balance the budget without dealing with mandatory spending programs like Medicare. We can’t save Medicare as we know it. We can save Medicare only if we change it.”