Congressional Republicans know they can’t stop a tax increase — but they’ll be damned if they let President Barack Obama get a debt limit increase for free.

The statutory ceiling on the U.S. debt, according to GOP lawmakers, staff and conservative interest groups, remains their strongest leverage in getting the president to come to the table with deeper spending cuts — now and in the future. And Obama, in his opening offer in fiscal cliff negotiations, proposed eliminating the need for Congress to affirmatively raise the ceiling, arguing politics should not affect whether the U.S. government meets its obligations.

But since the president made that offer, Republicans have dug in on the issue, becoming almost preoccupied by their need to preserve that point of leverage during Obama’s second term.

“It’s all our guys want to talk about,” one senior Senate Republican aide said. The aide noted that Democrats should not misjudge the GOP’s willingness to hold the debt limit hostage and that the party will be in no mood to compromise on the issue if Obama keeps insisting on a massive tax increase.

Sen. Rob Portman, former director of the Office of Management and Budget, defended using the debt limit as a legitimate tool to extract fiscal changes. The Ohio Republican said that’s the only thing that’s worked in recent decades to bring deficits under control — though he acknowledged it caused disruption in 2011. Portman appealed to Obama to come to the table, noting that the rest of his agenda — from immigration to spending ideas — will rely on having a working relationship with the GOP.

“He’s going to have to work with John Boehner, and he’s going to have to work with us,” Portman said.

So far, Speaker John A. Boehner, R-Ohio, has stuck to the Boehner rule — a demand that any debt increase be accompanied by an equal dollar amount of cuts or changes. Backing down from that position would be difficult for a restive GOP caucus already smarting from a showdown on taxes that it is likely to lose.

The political risks of repeating the 2011 debt ceiling strategy are obvious — the GOP’s Wall Street allies have no desire for an encore of the drama, which led to a downgrade in the U.S. credit rating by Standard & Poor’s. Federal Reserve Chairman Ben S. Bernanke warned Wednesday that business investment and hiring decisions were already being affected by the uncertainty surrounding the cliff talks, and he has said a debt limit default would be calamitous.

Plus, there would be no question which party allowed a default. And, this time around, Obama no longer has to worry about re-election. But there’s another risk for the Republican lawmakers — GOP primary voters and an array of outside groups such as the Club for Growth that are demanding members stick to their guns on cuts-for-debt.

“I hope their backs are very stiff,” said Club for Growth President Chris Chocola, who has been speaking with lawmakers on the Hill about the issue. “They seem to be focused on it, and that’s kind of the leverage they have.”

Obama’s comment that he wouldn’t “play” the game of negotiating with Republicans on the debt limit has buoyed the spirits of liberals in the Democratic Caucus, who doubt the GOP would have the ability to withstand the pressure a debt limit deadline would bring.

“The nation doesn’t have the stomach for it,” said Missouri Rep. Emanuel Cleaver II, chairman of the Congressional Black Caucus. “Everything we are hearing in the Democratic Caucus is he is holding fast to that, and we are pleased to hear it.”

Indeed, Democrats generally think the GOP is bluffing.

But Republicans say otherwise.

Sen. Patrick J. Toomey, R-Pa., brushed off the consequences of hitting the debt ceiling. “Is a temporary disruption worse than a full-blown debt crisis?” he asked. Toomey said Republicans should demand cuts in return for a debt increase and should be prepared to deal with the fallout. Toomey, a former Club for Growth president, said Obama could prioritize which government obligations to meet, thus preventing a default on interest on the debt.

That idea has been ripped by the administration — including Treasury Secretary Timothy F. Geithner — who has warned repeatedly that failing to meet other obligations for the first time in the nation’s history would still signal to the markets that the United States isn’t trustworthy. And Obama in 2011 cautioned that many other programs — such as Social Security payments — would be affected as well. Even under the prioritization scenario pitched by Toomey, almost $100 billion a month in bills would go unpaid. That is about 10 times the size of the sequester — automatic spending cuts that are scheduled to go into effect in January and that many lawmakers hope to avoid as part of any cliff deal. Additionally, $100 billion is enough to put a serious dent in the gross domestic product and crimp all manner of government services.

Democrats believe the consequences would be so dire that the GOP would effectively be shooting itself in the foot.

But Chocola said Republicans better stick by their threats. And he said Boehner should take a page from House Minority Leader Nancy Pelosi, D-Calif., and her successful push to enact health care reform when she was speaker.

“She knew it put her majority at risk,” he said. “She was willing to do that because she believed in it. Republicans can learn a lesson,” he said, noting that the health care overhaul remains the law of the land.

Chocola, like Toomey, said there is plenty of tax revenue coming in to prevent a default on the debt. “You may have some slow pay of our bills,” he said, but he added that interest payments are still a small sliver of the nation’s budget.

But White House Press Secretary Jay Carney continued to blister at the idea that Republicans might play chicken with default again as the “height of irresponsibility.”

“We cannot play this game because while it might be satisfying to those with highly partisan and ideological agendas, it’s not satisfying to the American people. … The president will not engage in that kind of brinksmanship. He just expects Congress to do its job,” Carney said.