The two sides also differ on the estate tax, extending unemployment benefits and how to address the need to raise the government's borrowing cap to prevent a first-ever U.S. default and a re-run of last year's debt crisis.
The White House was facing its own backlash, with labor, liberal and elderly advocacy groups mounting an organized campaign against any adjustments in cost-of-living for Social Security beneficiaries.
"President Obama and other Democrats campaigned saying Social Security doesn't affect the deficit," said Roger Hickey, co-director of the liberal Campaign for America's Future. "Social Security recipients are going to notice and they are either going to blame John Boehner or President Obama."
The change would reduce annual cost-of-living increases for beneficiaries of Social Security and other government programs. It also would push more people into higher tax brackets by making smaller annual adjustments to brackets.
The administration appeared confident that most Democrats would reluctantly vote for the idea in an attractive enough budget package.
White House spokesman Carney described the inclusion of the inflation adjustment as "a technical change" that was "not directed at one particular program." He also said that if instituted, the administration would ensure that the most vulnerable beneficiaries would not be affected.
"As part of an effort to find common ground, with the Republicans, the president has agreed to put this in his proposal," Carney said.