The Republicans' new approach would require even deeper cuts in social programs than they pushed last year. Liberals denounced those earlier plans as severe and unfair, and they say the new version would be worse.
The new commitments by House and Senate members stem from the ongoing dispute over raising the federal debt ceiling. The House voted Wednesday to postpone any showdown over the borrowing limit for three months. The Democratic-led Senate plans to endorse the idea, which the White House also supports.
That means the next big budget clash will occur in March. That's when major, across-the-board spending cuts both parties dislike them are scheduled to begin unless they are replaced by a different deficit-cutting technique.
It's possible that both parties will continue to find ways to postpone and minimize tough decisions on taming the deficit. But the new House and Senate endeavors could make such dodges more difficult. Voters, meanwhile, may get a clearer picture of the unpleasant choices they face.
"The American people will have a chance to compare the two approaches," said Rep. Trent Franks, R-Ariz., who wants deep spending cuts and no new taxes. The only way to shrink the government, he said, "is to choke the monster."
House Republicans now vow to balance the budget in 10 years without tax hikes. They say the methods might include reducing future benefits for Medicare, and possibly Social Security, for people now in their late 50s, rather than those 55 and younger.
The search for savings must "move over into entitlement spending" more so than before, said Rep. John Fleming, R-La.
Balancing the budget in 10 years without tax increases would require deep spending cuts, causing many analysts to view them as politically near-impossible. Such an effort would require cuts "that are a complete nonstarter for anybody probably even including House Republicans themselves," said Michael Ettlinger, an economist at the Democratic-leaning Center for American Progress.
But House conservatives say aiming for that goal is important. They demanded this week that GOP leaders agree to balance the budget in 10 years as the conservatives' price for supporting the three-month extension of federal borrowing powers.
House Democratic leader Nancy Pelosi called the new Republican goal "Ryan on steroids." She was referring to earlier Republican budgets drafted by House Budget Committee Chairman Paul Ryan, the party's 2012 vice presidential nominee.
Ryan's earlier plans would have changed Medicare into a voucher-like program with limited government contributions to health care for seniors. His plans, endorsed by most House Republicans, also would have given states full responsibility for running Medicaid the health care program for the poor with a reduced federal contribution.
Even with those changes, Ryan's 2011 and 2012 plans would not have balanced the federal budget for decades.
As part of the deal putting off the debt limit showdown, the Democratic-run Senate made concessions of its own: It agreed to debate and pass a budget for the first time in three years. That exercise will force senators to commit themselves to politically distasteful spending cuts and tax increases that they previously had avoided.
Democrats, including President Barack Obama, say new tax revenues must be part of any eventual bipartisan budget accord.
"As we go forward to reduce the deficit, we need growth and job creation, we need spending cuts, we need revenue," Pelosi said during Wednesday's House debate.
Democrats say new limits on tax breaks that mostly go to high-income households could generate billions of dollars in revenue.
But Republican leaders, noting that Democrats achieved a 10-year, $600 billion revenue hike as part of the "fiscal cliff" compromise earlier this month, say further tax hikes are off the table.
The Republicans' "no new taxes" mantra has clashed many times with Democrats' vows to protect government programs. The result, for years, has been deficit spending.
The latest House and Senate decisions will merely heighten that debate.
Democratic senators, no longer able to sidestep budget details, are certain to renew their push for tax increases to accompany further spending cuts. And House Republicans have made their deficit-reduction goals harder to achieve by promising to balance the budget in 10 years instead of, say, 30.