Republican lawmakers moved ahead Monday on a doomed plan to amend the U.S. Constitution to require a balanced federal budget, one day after President Obama met with the top two House GOP leaders in hopes of reaching a debt-limit agreement that could win approval from the hostile House.

House Republicans insist that a constitutional mandate is the only way to impose discipline on Washington’s spending binges and spare the country future dangerous political showdowns like the one now speeding the country to the brink of its first-ever default.

Beyond requiring that the budget be balanced each year, the so-called “cut, cap and balance” measure under House consideration would require that the constitutional provision include annual spending caps and a supermajority to approve tax increases.

That plan faces almost certain defeat in the Senate, where many lawmakers moved ahead on Monday with a compromise proposal.

The current timeline, according to aides in both parties, would call for the Senate to unveil its bipartisan plan later this week and begin to consider it Saturday. With a week of parliamentary hurdles to clear, the Senate could pass its bill by July 29, leaving the House just four days to consider whether to approve the plan before Aug. 2, when the country would no longer be able to pay its bills.

Democrats are watching the action in the House warily. They hope that the cut, cap and balance measure will convince House Republicans that they need to sign on to the Senate compromise.

Obama issued a cautionary threat Monday to veto the measure should it pass Congress. The White House said the bill would set up an “unacceptable choice” between a disastrous failure to lift the nation’s borrowing limit or passing a balanced budget amendment that would require draconian spending cuts in Medicare, Social Security and other federal programs.

“This is a measure designed to . . . duck responsibility, dodge obligations and dismantle, eventually, if signed into law, our social safety net,” White House spokesman Jay Carney told reporters at his daily briefing.

Behind the public bluster were signs that quiet work continues on finding a compromise.

Obama met Sunday with House Speaker John A. Boehner (R-Ohio) and Republican Leader Eric Cantor (Va.) to discuss the way forward. While aides provided no details about the meeting, Treasury Secretary Timothy F. Geithner said in an interview Monday that he is “absolutely” confident a deal will be reached.

“Despite what you hear, and this is a complicated place, Washington, people are moving closer together,” he told CNBC.

In the Senate, Majority Leader Harry M. Reid (D-Nev.) and Republican Leader Mitch McConnell (Ky.) have been making quiet progress on a last-resort deal to allow the debt ceiling to be raised in return for about $1.5 trillion in spending cuts over 10 years.

A new idea also came from Oklahoma Republican Sen. Tom Coburn, who released a 600-page plan that he says would achieve $9 trillion in deficit savings over the next decade through a combination of proposals long anathema to both parties, including far-reaching spending cuts, entitlement reform and increased tax revenue.

Coburn challenged those in his party not to see the elimination of special-interest tax breaks as a tax increase, while maintaining that he does not support any effort to increase rates.

“Tax expenditures are not tax cuts,” he said. “Tax expenditures are socialism and corporate welfare. Tax expenditures are increases on anyone who does not receive the benefit or can’t hire a lobbyist or special interest group to manipulate the code to their favor.”

The House bill is an updated version of the kind of balanced budget amendment Republicans have coveted for years.

A dramatic shrink in the size of government would come in two steps. First, Congress would impose an immediate legal cap on spending that matches the budget written by Rep. Paul Ryan (R-Wis.), which covers the next 10 years.

Next, the debt ceiling would rise only after Congress approves sending a constitutional amendment to the states for ratification. That would require two-thirds of both chambers of Congress voting for it — an outcome that no one believes will happen.

While the measure does not specify how strict the spending cap should be, past proposals have included caps at 18 percent of gross domestic product, down from the current 24 percent.

White House economic adviser Jason Furman told reporters on a conference call that if defense were exempted from cuts, that kind of cap would require slashing 12 percent from the rest of the federal budget, including Social Security and Medicare, resulting in a $2,000 reduction in benefits for the average Social Security recipient.

Not even House passage of the measure is assured — some Republicans plan to oppose any legislation to lift the debt ceiling. House Republican Whip Kevin McCarthy (R-Calif.) on Monday predicted victory, but added that the leadership was “still working it.”

Leading conservative advocacy groups insist that Republicans must steer the measure through Congress before they would find a debt-ceiling increase acceptable.

The influential Club for Growth announced Monday that it supports the House measure — and that it would hold to account Republicans who support the Senate-brokered deal instead.

“A vote’s insufficient,” Tim Phillips, president of the tea party-affiliated Americans for Prosperity said recently. “It’s one thing to simply schedule a vote and say we gave you a vote. It’s another thing for it be elevated to a point where it’s part of the deal.”

More on

The Fix: Why Republicans don’t fear default

2Chambers: Nearly 30 percent of House skips votes as debt-limit deadline looms

Poll: Little confidence in leaders to deal with debt issue

Lori Montgomery, Felicia Sonmez and David Nakamura contributed to this report.