House Speaker John A. Boehner’s new offer to allow higher tax rates in a fiscal cliff agreement includes a proposal to raise the eligibility age for Medicare and use a tougher measure for inflation adjustments for Social Security, according to sources familiar with the negotiations.

The changes are part of the substantial cost savings to domestic entitlement programs that Boehner, R-Ohio, is seeking in exchange for higher tax rates for annual income of more than $1 million.

The GOP offer would raise the eligibility age for Medicare benefits from the current 65 to 67, a source said. Boehner also wants increases in Social Security benefits calculated under “chained CPI,” a more dynamic method of figuring inflation that would lower annual adjustments.

The offer came during a week when Boehner and President Barack Obama met twice at the White House and spoke on the phone at least twice, including on Friday. Talks that had seemed at an impasse appeared to be getting more serious with back and forth offers aimed at an agreement to replace the tax increases and automatic spending cuts due to take effect at the start of next year. Aides for both parties said Obama had not accepted Boehner’s offer, although published reports said White House officials had characterized the offer that apparently was made Friday as progress.

But Democrats and groups backing the domestic programs have been marshaling forces in the past two weeks against such changes, saying they would hit the poor and the middle class the hardest.

Senate Majority Whip Richard J. Durbin, D-Ill., said last week he would not support an age increase without a guarantee of affordable coverage for the affected seniors. “Until we come up with gap coverage, I don’t think it’s a viable option,” he said.

Democrats also argue it would not generate much in cost savings because the greater medical expenses come for older beneficiaries. The Hudson Institute, an independent, conservative-leaning think tank, has said raising the age to 67 would generate about $125 billion in savings over 10 years.

For Republicans, the offer marks the first description of specific savings they would seek in entitlement programs after several weeks of demanding specifics from the White House on cuts the president would accept in domestic spending. The White House has countered that they have offered to trim some $340 billion from Medicare over a decade and take another $250 billion in unspecified reductions in entitlements.

And the White House and Republicans still can turn to an array of changes in entitlement programs that were part of the deficit reductions talks in the summer of 2011. Although the two sides never reached agreement on specifics, they discussed actions that could be part of a broader agreement. Those included such things as $53 billion in savings over a decade from tougher standards for Medigap, the popular program to cover gaps between Medicare and private insurance coverage.

Meantime, Democrats also say the Boehner offer on tax rates for the highest earners falls well short of deficit reduction targets.

A senior Democratic aide said the main problem with Boehner’s proposal on rates is that it would raise about $200 billion, not enough to meet Obama’s target for raising a total of $1.4 trillion in new tax revenue as part of a debt deals. Combined with Boehner’s offer to raise $800 billion in new revenue over a decade from a rewrite of the tax code, that would leave $1 trillion in tax revenue.

But for Democrats, the tax proposal also highlights past divisions in the party over the rates.

Boehner’s offer echoes past Democratic proposals for a millionaire’s surtax that have always been strongly opposed by the GOP and by many business trade groups in Washington. The latest proposal could drive a wedge in the Democratic caucus, as it resembles proposals that have drawn support from Democrats from states with high-income cities and suburban areas such as California, New York, Connecticut and Illinois. A number of Democrats from New York, for example, have pushed to raise the income ceiling for tax cut extensions to reflect higher average incomes in New York City and surrounding communities.

Paul M. Krawzak contributed to this report.