At the same time, President Barack Obama said the two sides are "making progress" after a series of back-channel conversations with Republicans like House Speaker John Boehner of Ohio and Majority Leader Eric Cantor of Virginia, with whom Obama met on Sunday morning.
Boehner called the veto threat "unfortunate," and in a statement Monday said Obama's threat "should make clear that the issue is not congressional inaction, but rather the President's unwillingness to cut spending and restrain the future growth of our government."
"If we are going to raise the debt limit and avoid default, the White House must be willing to demonstrate more courage than we have seen to date," Boehner said.
A Boehner aide said the "cut, cap and balance" plan is still "the best path forward" and declined to characterize the hush-hush talks with Obama. Another spokesman, Brendan Buck, said "there is nothing to report in terms of an agreement or progress."
The House set to pass the plan on Tuesday but it's sure to stall in the Senate, where majority Democrats say it would lead to decimating budget cuts and make it harder to pass tax increases on the wealthy. The White House called the bill an "empty political statement" that Obama would veto.
And even if the scheme could pass, there's no way Congress will adopt a balanced budget amendment, which requires a two-thirds vote in both House and Senate.
Tea party lawmakers are insisting on the effort to try to put their stamp on the debate over the so-called debt limit, and GOP leaders - lacking other ideas that might win a majority in the Republican dominated House - were quick to give their OK at a spirited closed-door meeting on Friday.
Major rating agencies such as Moody's and Standard and Poor's have signaled they're poised to lower the nation's coveted Triple-A credit ratings if no agreement is reached on the nation's debt.
On Monday, Moody's issued a report suggesting that the U.S. could eliminate the threat to its credit rating by removing the legal limit on the debt.
Tuesday's House vote on the cut, cap and balance plan comes after more than a week of White House talks with congressional leaders failed to produce a breakthrough. The tally will be held exactly two weeks before an Aug. 2 deadline to avoid a potentially devastating default on U.S. obligations like payments to bondholders and senior citizens receiving Social Security.
The plan doesn't actually include any spending cuts; rather, it endorses a spending cap on agency operating budgets for next year that the House is already working toward. The new plan promises to cut $35 billion more from so-called mandatory programs like farm subsidies, food stamps and Medicaid, the health program for the poor and disabled, but doesn't say where they would fall.
In a statement, the White House warned that such an approach could force significant cuts in education, research, Medicare and Social Security, and "put at risk the retirement security for tens of millions of Americans."
Obama spokesman Jay Carney derided the "cut, cap and balance" proposal as "duck, dodge and dismantle." He said behind-the-scenes work continues on measures that can actually pass both chambers of Congress, including a variation of a plan offered by Senate Republican leader Mitch McConnell to give Obama the power to raise the debt limit.
"Americans should not be concerned," Carney said. "We believe the debt ceiling will be raised. The United States will not default."
After the House exercise and a failed Senate vote on a balanced budget amendment to the Constitution, Plan B appears to be to have the Senate vote to give Obama sweeping power to order increases in the debt limit totaling $2.5 trillion over the coming year without approval by lawmakers.
The hope appears to be that after trying it their way, enough House Republicans will be able to stomach the emerging Senate plan to allow it to pass, though it's plain there will have to be plenty of Democrats voting for it as well to make up for dozens of unyielding lawmakers unwilling to abandon their tea party promises.
For their part, Senate Republicans are demanding a vote this week on a balanced budget constitutional amendment, though they have zero chance of winning the required 67 votes in a chamber where they control just 47.
Public opinion polls show that voters like the idea of a balanced budget, but the government faces such massive budget gaps - it now borrows more than 40 cents of every dollar it spends - that the cuts required to eliminate the deficit were too draconian for even the GOP-dominated House to endorse balancing the budget anytime soon. The House Republican budget still leaves deficits in the $400 billion range after 10 years.
The immediate issue is allowing the government to continue to borrow from investors and foreign countries like China to pay its bills - which include a $23 billion batch of Social Security payments set to go.
The report Moody's issued Monday said the need for the U.S. to raise its debt limit by enacting legislation creates the possibility of default that wouldn't exist otherwise. Removing the legal limit on the debt would eliminate the risk that threatens the government's credit rating, the report said. "We would reduce our assessment of event risk if the government changed its framework for managing government debt to lessen or eliminate that uncertainty," said the Moody's report.
With the deadline just over two weeks away and with a recent round of White House talks failing to generate a breakthrough, McConnell, R-Ky., the cagey leader of his party in the Senate, has proposed a plan that would allow Obama to automatically win a large enough increase in the debt to keep the government afloat until 2013 unless both House and Senate override him by veto-proof margins.
McConnell's plan has political advantages but has come under assault from many conservatives eager to take advantage of the current opportunity to use the need to lift the debt ceiling to force deficit cuts now. But Republicans refuse to consider any tax revenue increases demanded by Obama and Democrats to balance any budget package, and Democrats won't go along with significant cuts to benefits programs like Medicare and Medicaid unless tax increases on the wealthy are a part of the package.
That leaves lawmakers well short of the $2 trillion-plus in deficit cuts required to offset a debt increase that's big enough to solve the problem through next year's elections.
Majority Leader Harry Reid, D-Nev. wants to add a plan to create yet another deficit panel, comprised entirely of lawmakers and evenly divided between the two parties, that would try to break the deadlock by the end of the year.
It's also expected that a package of spending cuts, perhaps in the $1.5 trillion range over the coming decade, would be attached to the McConnell-Reid measure, perhaps in the House.
Carney said the dragged-out debt standoff at a time when both parties have reason to work together is a perfect example of "what Americans hate about Washington."
Associated Press writers Ben Feller and Erica Werner contributed to this story.