Washington ? The next step in the weeks-long saga over how to increase the government’s borrowing cap is to let House tea party forces try it their way.

A Republican “cut, cap and balance” plan set for a House vote Tuesday would condition a $2.4 trillion increase in the so-called debt limit on an immediate $100 billion-plus cut from next year’s budget and adoption by Congress of a constitutional amendment to require a balanced budget.

“Let’s let the American people decide,” said Rep. Jim Jordan, R-Ohio, on “Fox News Sunday.” “Do they want something common sense as cutting spending, capping the growth in government and requiring a balanced budget amendment to the Constitution?”

The idea appears to be to allow tea party-backed GOP lawmakers to have the run of Congress this week in hopes that they’ll ultimately be able to stomach a plan emerging in the Senate to give President Barack Obama sweeping power to order a $2.5 trillion increase in the debt limit without approval by Congress.

The cut, cap and balance plan, however, is a dead letter with Obama and in the Democratic-controlled Senate — as is a separate effort by Republicans in that chamber to adopt a balanced budget amendment. Amending the Constitution requires a two-thirds vote of both Houses, including 67 votes in the Senate, where Republicans control just 47.

“No one believes there are 67 votes for any version of that,” said Dick Durbin of Illinois, the No. 2 Senate Democrat, on CBS’ “Face The Nation.”

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 checks set to go out the day after an Aug. 2 deadline to avoid default.

With the deadline just more than two weeks away and with a recent round of White House talks failing to generate a breakthrough, Sen. Mitch 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.