Updated

Throwing down the gauntlet, Republican Sen. Jim DeMint threatened Monday to block a vote in Congress on raising the U.S. debt ceiling unless he wins a balanced-budget amendment to the Constitution.

The filibuster threat comes a day after Treasury Secretary Tim Geithner suggested Republican leaders had offered private assurances to the White House that they ultimately would vote to raise the $14.3 trillion ceiling, regardless of whether a deal is reached on long-term spending cuts.

Publicly, Republicans say they will demand spending cuts as a condition for supporting a hike in the debt ceiling. They stood by that claim following Geithner's comments, and DeMint took their demands a step further.

"I will oppose any attempt to vote to raise the limit on our $14 trillion debt until Congress passes the balanced-budget amendment," the South Carolina conservative said. He first made the remarks to McClatchy, which his office confirmed to Fox News.

A balanced-budget amendment would prohibit the U.S. government from running a deficit. Such a provision would take a two-thirds vote in Congress, in addition to ratification by the states.

White House spokesman Jay Carney did not address DeMint's threat directly during his briefing with reporters on Monday, but he did say that the "cleanliness" of the bill -- in other words, that there are no policy attachments to it -- is not an issue.

"The issue here is the debt ceiling has to be raised, and it cannot be held hostage to a process that is very complicated and difficult," he said. "We hope we will reach an agreement on deficit reduction -- a bipartisan agreement on deficit reduction within the time frame. We believe that's possible."

All GOP senators already have signed onto a balanced-budget amendment proposal, reviving a push from the mid-'90s -- when the House approved such an amendment, and the Senate fell one vote short of doing the same.

DeMint's demand, though, goes beyond those of other Republicans who say they just want to see a serious plan for closing the deficit as a condition for support on a debt-limit increase.

Sen. Tom Coburn, R-Okla,, said on "Fox News Sunday" that he needs to have "absolute certainty" a deficit reduction plan includes "critical changes."

"Unless we do that, there's no way I'll support it," Coburn said on the debt ceiling increase.

Sen. Rand Paul, R-Ky., on CNN's "State of the Union," said it's "yet to be determined" whether he would support a filibuster on the debt ceiling vote.

Rep. Chris Van Hollen, D-Md., though, said Congress should not "monkey around with the full faith and credit of the United States."

"Linking the two and saying you're only going to vote for the debt ceiling if something particular happens on deficit reduction I think is playing Russian roulette with, like, the fully loaded revolver," he said on "Fox News Sunday."

Geithner said repeatedly Sunday that lawmakers who want to take the country to the "brink" will bear the responsibility for the risk that creates.

He suggested that merely flirting with that edge would create a problem. But he said if Congress ultimately rejects an increase in the debt limit, it would trigger a crisis that makes that 2008 meltdown look tame. Geithner reiterated warnings that such a vote would force the government to halt benefits payments to seniors and veterans and would risk the government defaulting on its interest.