President Obama and lawmakers in both parties latched on to a new strategy for reducing the federal debt Tuesday, saying an emerging plan to save $3.7 trillion over the next decade could help break a political impasse over the debt limit and avert a U.S. default.

The proposal, crafted by a bipartisan group of senators known as the “Gang of Six,” calls for $500 billion in immediate savings and requires lawmakers in the coming months to cut agency spending, overhaul Social Security and Medicare, and rewrite the tax code to generate more than $1 trillion in fresh revenue.

In the works since January, the plan became public Tuesday, just as it was becoming apparent that the leading option for raising the federal debt limit faces bleak prospects in the House. With the Treasury expected to start running out of cash in two weeks, House Republicans are openly hostile to a Senate-led plan that would authorize additional borrowing but kick the hardest decisions — such as whether to raise taxes or cut entitlement programs — to a special legislative panel.

The Gang of Six framework, by contrast, would force lawmakers to make some of those tough decisions now. The urgency of the situation was underscored Tuesday when a major credit-rating company said it could downgrade five states, including Maryland and Virginia, in addition to the federal government if the United States defaults on its obligations.

As many policymakers were turning their attention to the new strategy, the Republican-controlled House forged ahead Tuesday with another approach to the debt crisis, voting to sharply cut spending and tie an increase in the debt limit to the eventual adoption of a balanced-budget amendment. The measure is unlikely to pass the Senate, and Obama has promised to veto the bill if it does.

The new plan emerging from the Senate may have better prospects. House Speaker John A. Boehner (R-Ohio) said through a spokesman that the Gang of Six plan “shares many similarities with” the far-reaching strategy he had been pursuing with Obama this month but abandoned amid GOP opposition to fresh revenue. The president agreed, and he hailed Republican openness to a plan that includes new taxes as “a very significant step.”

“It’s time to get down to the business of actually solving this problem. And I think we now are seeing the potential for a bipartisan consensus around what that would take,” Obama told reporters during a quick appearance in the White House briefing room.

Obama summoned congressional leaders to the White House again Wednesday to “start talking turkey” about the shape of a compromise package that could pass by Aug. 2.

“The problem we have now is we’re in the eleventh hour,” Obama said, “and we don’t have a lot more time left.”

Neither Obama nor Boehner embraced the specific details of the Gang of Six proposal, and it was not immediately clear how the strategy might influence debt-limit negotiations. House Majority Leader Eric Cantor (R-Va.) said in a statement that the plan contains “some constructive ideas for dealing with our debt,” but he objected to the revenue goals, arguing that “a tax increase is the wrong policy to pursue with so many Americans out of work.”

Senate leaders, who have long been skeptical of the Gang of Six were notably cool to the proposal. Minority Leader Mitch McConnell (R-Ky.) offered no opinion, while Majority Leader Harry M. Reid (D-Nev.) highlighted the procedural obstacles to shifting legislative strategies so late in the game.

Reid, who has been adamant in his opposition to cutting Medicare and Social Security benefits as part of the debt-limit negotiations, acknowledged that he does not support the proposal.

“For me to say I love this bill that other people have worked on for months and months, I can’t do that,” Reid told reporters, adding, “Remember, we have only 13 days.”

The ticking clock is a major impediment to pursuing the Gang of Six strategy, which has yet to be drafted in legislative form or examined by congressional budget analysts. Opponents ripped into the sketchy details handed out at a morning briefing attended by about half the Senate, arguing that printed summaries identify nowhere near $3.7 trillion in savings.

Still, the plan was attracting interest among House Republicans eager for bold action to restrain borrowing. Rep. Steven C. LaTourette (R-Ohio), a close Boehner ally, said, “Anything that moves toward the grand bargain that the speaker and the president have been working on is great.”

And freshman Rep. Charles F. Bass (R-N.H.) called the proposal “a move in the right direction.”

“There’s tax reform, there’s significant spending reduction, and there’s also a debt-ceiling increase,” Bass said. “This is a positive development, and I’m going to be very open-minded about it.”

In the Senate, the plan received a warm response, both during the invitation-only morning session and during separate luncheon briefings for Democrats and Republicans. Sen. Lamar Alexander (Tenn.), the No. 3 Senate Republican, offered strong support. And Sen. Tom Coburn (R-Okla.), a leader in the GOP on budget issues, backed the plan as well, after dropping out of the Gang of Six in mid-May.

“There was palpable relief from folks in that room this morning. Like they were saying, ‘Here’s something we can actually be for,’ ” said Sen. Mark R. Warner (D-Va.), a Gang of Six member who has relentlessly pressed his colleagues to reach consensus.

Now that the plan is in the open, Warner said, “it’s put-up-or-shut-up time — for all of us.”

The Gang of Six was formed in the wake of Obama’s fiscal commission, an 18-member panel whose ideas sparked broad interest. Reluctant to let that effort die, Warner and Sen. Saxby Chambliss (R-Ga.) joined forces with four senators who served on the commission to try to draft legislation to bring its recommendations to a vote.

The newly released framework relies heavily on the fiscal commission’s work, calling for deep cuts at government agencies, including the Pentagon; significant reductions to Medicare and Medicaid; and a plan to make Social Security solvent.

It also calls for raising more than $1 trillion over the next decade by reducing a variety of popular tax breaks and deductions, including breaks for home mortgage interest and employer-provided health care. While some of those savings would be dedicated to debt reduction, the rest would go toward lowering tax rates for everyone, with top individual and corporate rates dropping to at least 29 percent, down from 35 percent.

The work on taxes and entitlements would be done largely by existing legislative committees that would be given targets for achieving savings over the next six months. If any committees failed to act, 10 senators — five from each party — could step in to offer their own proposal.

Senate Budget Committee Chairman Kent Conrad (D-N.D.), a member of the group, said Senate leaders were looking at ways to add Gang of Six goals to legislation already in development to raise the debt ceiling.

But “the first test,” Conrad said, is getting 60 senators to sign a letter saying, “This is a direction that deserves support.”

The emergence of the Gang of Six proposal overshadowed political sparring in the House on Tuesday over a GOP plan to cut spending by $111 billion next year, impose strict caps over the next decade and raise the $14.3 trillion debt limit only after Congress approves a constitutional amendment requiring a balanced budget.

That measure, which passed 234 to 190, has virtually no chance of approval in the Senate. But lawmakers in both parties said the debate could prove useful in the quest for a final deal because it allows conservatives to go on record with their preferred plan before considering a less palatable alternative.

“I think there’s recognition that it’s not going to be signed into law,” Bass said of the measure known as “cut, cap, balance.” “But it’s an important step [that] allows people like me, who do support a balanced-budget amendment, an opportunity to vote on that.”

Staff writer Paul Kane contributed to this report.