Even as President Obama and congressional leaders focus on a fallback plan to lift the nation’s debt ceiling, top Democrats and Republicans have begun to map a new way to craft the same sort of ambitious deficit-cutting plan they abandoned last week.

As part of the deal being discussed to raise the debt ceiling, leaders on Capitol Hill are forming an especially powerful congressional committee that would be charged with drawing up a new “grand bargain,” possibly by the end of the year.

Key elements for a big deal remain in place. Obama has been clear that he wants one and has started making the case to skeptical factions of his own party that getting the nation’s fiscal house in order is in their best interest. House Speaker John A. Boehner (R-Ohio) also remains committed to an ambitious plan, having told his troops that he didn’t become speaker to do small things. And, perhaps most critically, the markets are demanding it. The credit rating agency Standard & Poor’s says Washington must agree to reduce the debt by $4 trillion over 10 years to avert a downgrade.

“We cannot as a country fail to deal with the debt threat,” said Senate Budget Committee Chairman Kent Conrad (D-N.D.), one of the bipartisan “Gang of Six” senators who tried to reach an agreement in recent months. “Every serious economic analysis tells us we’ve reached the danger zone. And just kicking the can down the road? That can’t be. We’re better than that. We’ve got to be better than that.”

But hopes for a grand resolution in coming months face the same question that hangs over the current crisis: whether tea-party-aligned conservatives in Congress who forced the debt-ceiling showdown will provide the necessary votes for an eventual major deal, even if it includes new taxes.

The Gang of Six — which tried to come up with its own plan — disbanded last week after failing to reach agreement on how to cut spending and raise taxes. Meanwhile, the big deal pursued by Obama and Boehner faltered amid criticism from congressional Republicans opposed to additional revenue.

House Majority Leader Eric Cantor (R-Va.), who has emerged as the leader of this contingent, has argued against such a deal. But his views may be shifting along with those of some rank-and-file House Republicans whose imaginations have been captivated by the idea of slicing as much as $5 trillion out of the federal budget over the next decade.

In a caucus meeting last week, some freshmen wanted to know whether they could slice that much out of the budget in the next two years, GOP aides said. (Answer: no. The entire federal government is expected to spend about $3.6 trillion this year.)

In public at least, conservatives are maintaining that the answer is to “cut, cap and balance” — passing a balanced-budget amendment that would cap federal spending at 18 percent of the nation’s gross domestic product, down from its current 24 percent.

The House is expected to vote Tuesday on such an amendment, but it has scant odds of getting the needed supermajority in the Senate. Democrats say an 18 percent cap in a country with an aging population and rising health-care costs would lead to ruinous cuts.

But conservatives said Saturday that they are holding out for the amendment and are not ready to accept a stopgap measure being put together by Senate leaders Mitch McConnell (R-Ky.) and Harry M. Reid (D-Nev.) that would raise the debt ceiling before the nation hits its borrowing limit Aug. 2.

“I didn’t get elected to punt this problem down the road another six months,” said Rep. Jason Chaffetz (R-Utah). “We are the body, we are the commission to make these tough decisions. . . . Guys like me are not coming along. We’re not going along just to get along.”

Rep. Allen B. West (R-Fla.) was equally blunt. “The quote-unquote McConnell-Reid plan is no plan. That’s the acquiescence of the responsibilities of our Congress,” he said. “It’s nothing but the typical D.C. two-step, and I’m not going to be part of that.”

Under the stopgap plan, Congress would allow Obama to raise the debt ceiling in three increments totalling $2.5 trillion over the next year. Each time, Congress would vote on a resolution of disapproval, allowing Republicans to blame the increases on Obama.

To get backing from House Republicans, McConnell and Reid are adding $1.5 trillion in spending cuts. And they are drawing up the committee, with six lawmakers from each party, which would report by the end of the year.

The committee would resemble the fiscal commission chaired last year by Alan Simpson and Erskine Bowles, minus presidential appointees. But its path would be easier: The panel will require only a simple majority to report a plan to Congress, it would be protected from Senate filibuster and it would not be subject to amendment, similar to the system used for closing military bases.

The same lawmakers appointed to the fiscal commission may not be appointed to the new committee. Aides said Reid, for one, is likely to make different choices. But Sen. Richard J. Durbin (D-Ill.) said there is little doubt where the committee will start.

“We all keep coming back to the same basic parameters. The Bowles-Simpson plan really laid out how you reach $4 trillion. And there aren’t a lot of things they didn’t consider,” Durbin said. “There are only so many moving parts here.”

The commission recommended saving $3.8 trillion by raising the retirement age for Social Security, slashing spending across government and wiping out more than $100 billion a year in popular tax breaks, including the tax deduction for mortgage interest and the tax-free treatment of employer-provided health insurance. It recommended larger Pentagon cuts and revenue increases than the White House sought this month.

The Gang of Six arrived at about $4.65 trillion in savings, according to Conrad, but disbanded after failing to get buy-in from its Republican members, especially Sen. Tom Coburn (Okla.). He plans to offer his own proposal Monday, the most ambitious yet, with $9.5 trillion in savings.

Skeptics of a Simpson-Bowles type of grand bargain have, until recently, included Reid and McConnell. Reid has been outspoken about the need to protect Social Security benefits. He is also eager to avoid cuts to Medicare that would undermine Democrats’ ability to campaign against the House GOP’s plan to eventually replace Medicare with a system of subsidies to buy private coverage.

Obama countered last week that Democrats should want a major fiscal deal, because it would make it easier to win approval for spending on their priorities in the next few years.

“If you care about making investments in our kids and making investments in our infrastructure and making investments in basic research,” he said, “then you should want our fiscal house in order so that every time we propose a new initiative somebody doesn’t just throw up their hands and say, ‘Ah, more big spending, more government.’ ”

Two conservative Senate Republicans said Saturday that they are highly doubtful that the new committee would produce a real solution, which is why they are holding out for a balanced-budget amendment.

Sen. Mike Lee (Utah) said, “The question isn’t so much whether [the committee] could come up with something; it’s whether what it comes up with is something Congress could follow” over time. Sen. Patrick J. Toomey (Pa.) said the “abdication of responsibility” to a committee was “disturbing.”

“I understand what Senator McConnell is trying to do here,” Toomey said. “But at the end of the day, this arrangement is going to authorize the president to issue another $2.5 trillion in debt, and I can’t be part of that.”