WASHINGTON — Senate Democrats offered a new proposal to extend the payroll tax holiday, a strategy designed to attract Republicans who have been cool to continuing a tax break for working Americans that expires Dec. 31.

President Barack Obama pressed the case Monday after Senate Republicans revolted against earlier plans last week and House Republicans panned the tax-cut proposals.

House Speaker John A. Boehner, R-Ohio, also is crafting a new GOP alternative designed to attract votes from reluctant lawmakers. Congress has just a few weeks left to strike a compromise. The break is worth about $1,000 annually to the average family.

“As soon as this year ends, so does that tax cut. There aren’t many folks, either in the middle class or those trying to get into the middle class, who can afford to give up $1,000 — not right now,” Obama said in remarks from the White House. “That’s why Congress must act.”

The new proposal from Senate Majority Leader Harry Reid, D-Nev., seeks to address GOP concerns. It would pay for the cost of the tax break with a combination of GOP-backed proposals to increase the fees that Fannie Mae and Freddie Mac charge mortgage lenders, as well as a surtax on those earning beyond $1 million a year.

Those revenue streams would generate almost $185 billion, and also replenish the Social Security trust fund. The tax holiday reduces the 6.2 percent tax workers pay toward Social Security to 4.2 percent. But the new revenue offsets that loss, and is designed to ease concerns from Republicans, and some Democrats, about the impact of the tax holiday on the retirement fund.

In keeping with Obama’s original proposal, the Democratic bill would increase the payroll tax break, reducing the tax to 3.1 percent to boost the tax savings for 2012 to an average of $1,500, a proposal that will certainly limit GOP support.

In a gesture to the GOP, Democrats dropped a provision sought by Obama that would have reduced the payroll tax for companies that hire unemployed workers. Democrats also tacked on a GOP proposal to prevent millionaires from receiving food stamps or unemployment benefits.

“Republicans need to be prepared to meet us part way,” Reid said Monday.

Republicans are unlikely to embrace this latest proposal — they have consistently opposed Democrats’ calls for a tax increase on millionaires — and immediately portrayed this week’s Senate vote as more designed to score political points than resolve the standoff. The proposal is likely to fail.

Boehner hopes to build support for extending the tax break by proposing a larger legislative package that includes an extension of long-term unemployment benefits and other routine tax measures that also expire at year’s end.

In a move to attract business-friendly lawmakers, Boehner wants to tack on a provision that would advance development of the Keystone XL pipeline, a controversial oil project that Republicans say will create jobs but critics argue poses environmental problems along the pipeline’s proposed route. Obama has shelved a decision on the pipeline until after the 2012 election.

Boehner is expected to unveil his plan this week.

A coalition of environmental groups, including the Sierra Club and Natural Resources Defense Council, wrote to congressional leaders Monday urging them to avoid linking the issues.

“The environmental issues that might be folded into this package are significant in their own right, and should not be slipped into an unrelated bill, limiting scrutiny and debate, in an effort to extort votes to roll back environmental and health protections,” the groups wrote.

Even though Boehner and Sen. Mitch McConnell of Kentucky, the Senate GOP leader, have insisted that Republicans support continuing the payroll tax holiday into 2012, their lawmakers are not so certain.

Arizona Sen. Jon Kyl, the No. 2 Republican in the Senate, said Monday there are “reasons to be very skeptical about extending this payroll tax holiday.”

Republicans have concerns about continuing a tax break that they say does little to stimulate the economy. They also want the costs of the tax cut offset with budget cuts elsewhere, rather than relying on a tax increase.

Economists say the tax break boosts the economy by putting money in pockets of those who are likely to spend it, and warn that failing to continue it would slice into the nation’s economic growth for 2012.