President Barack Obama accepted a move by Senate Democrats to scale back his Social Security payroll tax cut extension on Monday, then prodded Republicans to support it despite a requirement for the very wealthy to pay more taxes.

He spoke as Senate Democrats unveiled revisions that cut the cost of the administration’s proposal by one-third, to an estimated $179 billion. As rewritten, it deepens the current Social Security payroll tax cut and extends it until the end of 2012, but jettisons Obama’s request to give businesses relief at the same time.

Republicans, while not formally rejecting the latest proposal, dismissed it as an example of Democrats issuing a political offering they had not consulted with Republicans on, suggesting any real compromise would come later this week or next week.

“Senate Democrats say they have a credible ‘compromise’ proposal, which presumably does not include job-killing tax hikes or phony war savings. We look forward to reviewing it,” said Michael Steel, a spokesman for House Speaker John Boehner, R-Ohio.

Advanced by Sen. Bob Casey, D-Pa., the revised proposal scales back the surtax on seven-figure earners that Democrats had originally proposed to cover the bill’s entire cost, from 3.25 percent to 1.9 percent.

Casey said the proposal also adopts deficit-reducing measures from the recent “supercommittee” negotiations, raising $38.1 billion by hiking fees that Fannie Mae and Freddie Mac charge mortgage lenders.

And it would pare down the surtax on income over $1 million from 3.25 percent to 1.9 percent and make the surtax temporary, expiring after 10 years. Casey said this surtax would affect only 0.2 percent of taxpayers, who have an average annual income of nearly $3 million, and would take effect in 2013.

Also included is a GOP proposal from last week to make sure millionaires don’t receive unemployment benefits or food stamps.

Obama also called on lawmakers to renew a program of extended unemployment benefits due to expire on Dec. 31. He said the checks, which kick in after six months of joblessness, are “the last line of defense between hardship and catastrophe” for some victims of the recession and a painfully slow recovery.

Ultimately, both sides have said they believe a negotiated compromise between Democrats and Republicans in both the House and Senate will be necessary to find a way to pay for extending the tax holiday acceptable to both parties.

Still, Senate Democrats believe they have significant leverage on the issue.

They think the tax cut for workers is widely popular, as is their idea to pay for it with a surtax on millionaires. Already, one Republican, Maine Sen. Susan Collins, voted with 50 Democrats last week to extend the tax cut. Democrats think more might come along rather than risk repeatedly voting against a tax cut.

The president made his remarks at the White House as Republicans and Democrats in Congress said a holiday-season package was beginning to come into focus that could cost $180 billion or more over a decade. Elements include not only the payroll tax cut and unemployment benefit renewals, but also a provision to avert a threatened 27 percent reduction in fees to doctors who treat Medicare patients.

While there are differences over the details of the three principal components — many Republicans are reluctant to extend the tax cut — there is at least as much disagreement between senior lawmakers in the two political parties over ways to cover the cost so deficits don’t rise.

House Republicans are drafting legislation to extend an existing pay freeze for federal workers as partial payment for the tax cut and unemployment benefits. Other cost-savers are expected to include a proposal Obama advanced earlier this year to raise pension costs for federal employees, officials said. The bill may also include another presidential recommendation, this one for a surcharge on Medigap policies purchased by future Medicare recipients.

Officials said that to offset the two-year, $38 billion price tag of the Medicare provision, House Republicans want to cut funds from the year-old health care legislation that stands as Obama’s signature domestic policy accomplishment. Some Democrats want instead to count defense funds approved but unspent for the wars in Iraq and Afghanistan — a proposal that many GOP lawmakers deem an accounting gimmick.

The Medicare proposal enjoys strong popularity among lawmakers in both parties. House Republican leaders signaled last week they intend to include it in the overall package as a sweetener for members of the party’s rank and file who are unhappy at the prospect of extending the payroll tax cut.

GOP critics say there is no evidence the current tax cut has helped create jobs, and also say they fear the impact of a renewal on the deficit and on the fund that pays Social Security benefits. A majority of Republican senators voted last week against a plan backed by their own leadership to extend the cut.

But Obama noted House Speaker John Boehner, R-Ohio, has said that the renewal would help the economy, and said the party’s Senate leaders had made similar comments.

“I couldn’t agree more. And I hope that the rest of their Republican colleagues come around and join Democrats to pass these tax cuts and put money back into the pockets of working Americans,” the president said.

Obama also added, “I know many Republicans have sworn an oath never to raise taxes as long as they live. How could it be that the only time there’s a catch is when it comes to raising taxes on middle-class families? How can you fight tooth and nail to protect high-end tax breaks for the wealthiest Americans, and yet barely lift a finger to prevent taxes going up for 160 million Americans who really need the help?”

The Associated Press and Washington Post contributed to this article.