Senate Democrats unveiled yet another plan Monday to extend a popular payroll tax holiday, but the political theater surrounding their move — including an appearance by President Barack Obama in the White House briefing room — seemed designed to score a public relations win and increase pressure on the GOP.

In a concession to Republicans, Democrats dropped a proposed payroll tax break for employers but continued to insist on deeper cuts on the amount employees pay. They also amended their proposal on how to pay for the legislation, adding bipartisan spending cuts as well as means-testing of unemployment and food stamps benefits similar to those offered in a Senate Republican payroll tax cut bill last week.

But Democrats also included a more modest millionaires surtax, a provision they’ve been trying to enact into law for months but one that Republicans have repeatedly blocked. Adding the millionaires surtax yet again, Republicans say, shows that the Democrats’ new plan is more about political gains — especially on a measure that is likely to command bipartisan support in the end.

Republican and Democratic aides said the payroll tax cut, which is set to expire Jan. 1, will be extended by Congress in an end-of-the-year catch-all package. But that measure — which is likely to include extensions of unemployment benefits and increased Medicare payments to doctors — will probably have to originate in the Republican-controlled House.

The exact composition of that package, and how it might be paid for, is still very much in flux. And lawmakers are hoping the looming Christmas holiday will force their opponents’ hands and prevent either side from tacking on sweeteners for their conferences that could jeopardize final passage.

In the interim, Democrats appear to be relishing the opportunity to demagogue the issue.

On Monday, Obama arrived in the briefing room to deliver a statement to the media that liberally took shots at Republicans.

“Virtually every single Republican voted against that tax cut. Now, I know many Republicans have sworn an oath never to raise taxes as long as they live. How can it be that the only time there’s a catch is when it comes to raising taxes on middle-class families?” Obama said. “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?”

House GOP leaders have been working to find a combination of unemployment insurance, the Medicare “doc fix” bill and a payroll tax cut extension that can pass muster with their conservative membership.

“Our Members overwhelmingly don’t want to do payroll, so it’s going to be a really big lift,” a GOP leadership aide said Monday, explaining that Speaker John Boehner’s (R-Ohio) plan hinges on inclusion of a handful of GOP deregulation and jobs bills as sweeteners.

Specifically, leadership has considered including a bill to reduce Clean Air Act regulations for boilers and legislation addressing the Keystone XL oil pipeline controversy. They also would include unspecified changes to how the unemployment program operates.

“You have a lot of really good things that are loaded” into the unemployment portion of the package and the deregulatory provisions that are designed to make fiscal conservatives drop their opposition to the payroll extension, the aide said.

But while GOP leaders will need those enticements to pass a package in the House, they are essentially dead on arrival in the Senate, where the boiler air rule and Keystone pipeline language will almost certainly be eliminated. Changes to the unemployment benefits system are also likely to become a flash point.

Democrats and Republicans in the Senate seem to be banking on the Christmas calendar working as a hammer to force a recalcitrant GOP to ultimately vote for a package that would not include those provisions.

But House Republican leadership aides warned that may be wishful thinking.

“The Senate needs to understand what we’re dealing with here,” a second leadership aide said, explaining that stripping any sweeteners from the bill is “going to be a problem.”

How much of a problem, however, remains unclear. Boehner has had to go through similar machinations with his Conference all year long, allowing conservatives to vent their frustration over the pace of spending reductions before bringing the bulk of his Conference into line and passing a compromise that, in the end, mostly favors the GOP.

But the number of Republicans unwilling to budge has continued to grow, particularly since this summer’s debt ceiling vote. And with elections looming large, Boehner could be sorely tested to bring 218 Republicans to the table for a final deal that can also pass the Senate.

The Speaker also has been very candid with his caucus about the importance of extending the payroll tax holiday and unemployment insurance benefits. For Congress to fail to do so right before Christmas could be a political nightmare for Republicans, particularly if Democrats are successful in pinning the blame on them.

For months, Obama has been campaigning against Congressional dysfunction. Republicans are quick to point out that if Democrats continue to insist on the millionaires surtax, they are more interested in helping the White House’s message than extending the law. Democrats say they want final passage but are happy to play out this political battle right before the holidays.

On Monday, Majority Leader Harry Reid (D-Nev.) was adamant on the Senate floor.

“If Republicans block this proposal, raising taxes on American families by $1,000 next month will have an immediate impact on our economy,” Reid said. “It will halt very singularly our still-fragile recovery in its tracks and drag us back into a recession.”