Senate Democrats on Monday unveiled a scaled-back version of a bill to extend the payroll tax holiday, but one that still pays for the cuts by taxing millionaires at a higher rate that Republicans have opposed.

At the White House, President Obama urged congressional Republicans to join Democrats in approving the extension, arguing that it is urgently needed to prevent what would amount to a tax hike on middle-class families next year.

The latest version of the Democrats’ bill was offered Monday by Sen. Robert P. Casey Jr. (D-Pa.), setting up a vote expected later this week. It comes as the two parties race to find a compromise to extend the one-year tax break for workers, which expires Dec. 31. Without a deal, taxes will rise on the average family by about $1,000 next year.

In an appearance in the White House briefing room, Obama waded into the latest skirmish in Congress, where GOP lawmakers are resisting Democratic proposals to pay for the plan by imposing a tax surcharge on people earning more than $1 million a year.

“My message to the Congress is this: keep your word to the American people and don’t raise taxes on them right now,” Obama said. “Now is not the time to step on the brakes. Now is the time to step on the gas. Now is the time to keep growing the economy, to keep creating jobs, to keep giving working Americans the boost that they need.”

“Americans overwhelmingly support our proposal to ask millionaires and billionaires to pay their fair share to help this country thrive,” said Senate Majority Leader Harry M. Reid (D-Nev.). “Republicans in Congress dismiss it at their peril.”

Casey said his compromise proposal attempts to enlist Republicans by trimming the cost of the package by roughly a third and using “bipartisan ideas” to pay for it.

He said in a statement that the compromise would no longer provide any tax break for employers, cutting the package to $185 billion from $265 billion. He said it would still cut in half (from 6.2 percent to 3.1 percent) the payroll tax paid by employees and the self-employed on their wages and salaries in 2012, giving the average family nearly $1,500 in additional take-home pay.

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.

The payroll tax, which funds Social Security, previously stood at 12.4 percent and was split between workers and their employers.

Economists think that extending the tax holiday could help stimulate the still-ailing economy by leaving more cash in the hands of consumers. But some Republicans would prefer a comprehensive rewrite of the tax code to the temporary tax break. And members of both parties fear undermining the solvency of Social Security with repeated holidays.

Senate Democrats failed to get the 60 votes necessary last week to advance a broader version of the bill that would have expanded the 2 percent cut offered to workers to 3.1 percent and, for the first time, offered some employers a 2 percent cut as well.

Obama said that renewing the payroll tax cut is “important for the economy as a whole” because it will spur spending and hiring and help families pay their bills. But he noted that virtually every Senate Republican voted against his proposal to expand the payroll tax cuts, which he said would have given a typical working family a cut of about $1,500 next year.

“Now, I know many Republicans have sworn an oath never to raise taxes as long as they live,” he said. “How can it be 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? It doesn’t make sense.”

But the inclusion of the tax on millionaires probably kills any chance for the new bill’s passage and is instead a sign that both parties are still working to score political points on the issue.

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.

A Senate Democratic aide said congressional Republicans are the only ones who don’t consider a millionaires surtax a “credible idea.”

“A majority of Republicans in America support that idea, and 75 percent of people overall do so,” he said. “Maybe this is a case where Republicans need to move in the direction of the American people instead of Democrats moving in the direction of the Republicans.”

And their hand has been strengthened by deep Republican divisions on the issue. A majority of Senate Republicans last week voted against a competing GOP-authored proposal that would have extended the tax cut but paid for it by imposing a three-year pay freeze for federal workers and shrinking for the federal workforce.

Republicans, while not formally rejecting the latest Reid proposal, dismissed it as an example of Democrats issuing a political offering that 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 A. Boehner (R-Ohio).

House Republicans are trying to come up with their own alternative, hoping to overcome objections from many rank-and-file Republicans who have resisted supporting the payroll tax holiday as ineffective at spurring hiring.

Boehner has floated the idea of attaching to the payroll extension other conservative proposals, such as a bill that would mandate a speedy decision from the State Department on a proposed energy pipeline from Canada to the Gulf Coast.

“We are continuing to work with our members on legislation to extend the current payroll tax holiday, protect Social Security, reform unemployment insurance and cut government spending,” Steel said.