President Obama opened the door Friday to a less ambitious compromise to the looming debt-ceiling crisis than the “grand bargain” he has been pushing for, acknowledging that other alternatives might be needed to win approval in time to avoid a federal default.

After two weeks of urging Congress to embrace his plan to shave $4 trillion off the projected national debt, Obama said other options must be on the negotiating table with the Aug. 2 deadline just days away. Those include a complex proposal that Senate negotiators were working on to find smaller savings and establish a new framework for tax and entitlement reform.

“If we can’t do the biggest deal possible, then let’s still be ambitious. Let’s still try to at least get a down payment on deficit reduction,” Obama said at a White House news conference.

White House Chief of Staff William M. Daley and Treasury Secretary Timothy F. Geithner met Friday in the Capitol with House Speaker John A. Boehner (R-Ohio) and House Majority Leader Eric Cantor (R-Va.). According to a Democratic lawmaker and Democratic and GOP aides briefed on the meeting, the officials discussed whether some portions of the “big deal” — which originally included major overhauls to the tax code and entitlement benefits — could be included in the Senate plan. GOP leaders rejected Obama’s previous offering because it calls for a net increase in tax revenue approaching $1 trillion.

But most congressional and administration officials focused instead on drafting a Senate-driven deal that could win bipartisan support in both chambers, with first steps possibly coming in the middle of next week.

The Senate effort, led by Majority Leader Harry M. Reid (D-Nev.) and Minority Leader Mitch McConnell (R-Ky.), would allow Obama to increase the debt limit three times between now and the end of next year. Congress could block the president only with a two-thirds majority.

A special House-Senate committee would also be formed to propose potentially far-reaching reforms to the tax code and entitlement programs, such as Social Security and Medicare. Eventually, aides said, the House would amend that package to include nearly $1.5 trillion in spending cuts over the next 10 years.

If the entire package comes together, it will also cover budget levels for the next two fiscal years that provide immediate savings — eliminating almost any chance of a federal government shutdown.

To clear all the parliamentary hurdles and enact the legislation by early August, congressional officials expect Reid to introduce legislation on the Senate package by Wednesday night.

“We are obviously running out of time,” Obama said Friday. “And so what I’ve said to the members of Congress is that you need, over the next 24 to 36 hours, to give me some sense of what your plan is to get the debt ceiling raised through whatever mechanisms they can think about.”

In the House, however, dozens of conservative Republicans have vowed not to support any increase in the debt limit unless it is accompanied by legislation — known as the Cut, Cap, Balance Act — that sets strict caps on future spending and sends a constitutional amendment requiring a balanced budget to the states for ratification.

“I am not prepared at this point to pick winners or losers,” said Boehner, who has been appealing to this conservative bloc. “Senator McConnell pointed out that his plan was being put on the table as a last-ditch effort. We’re far from the time for a last-ditch effort.”

The next move in the House will be a vote next week on the Cut, Cap, Balance Act, something of keen interest to Majority Whip Kevin McCarthy (R-Calif.). McCarthy, the lead GOP candidate recruiter in the 2010 campaign, is the leadership’s closest ally to the critical bloc of 87 freshman Republicans, and he considers those lawmakers critical to getting some form of balanced budget amendment approved along with the debt ceiling increase.

“They are both building for the perfect storm,” McCarthy said in an interview this week. “Every time you turn on the news and they’re talking more about the debt of America and they’re talking about the crisis of getting something done, the balanced budget amendment gets stronger.”

Nearly 40 House Republicans, including 20 freshmen, have signed a pledge that they will not vote to approve a debt-ceiling increase unless it comes with the balanced budget amendment. Democrats oppose the amendment because it would rule out any tax hikes to balance the budget unless approved by a super-majority.

“The simple truth is that we’ve already heard all the reasons why we should cut spending, cap spending and pass a balanced budget amendment to the Constitution. We’ve heard them from folks on Main Street, from folks who call our offices and from folks at our town halls. Now, as we once again find ourselves at the eleventh hour, it’s time for political leaders to join our efforts to balance the budget,” Rep. Mick Mulvaney (R-S.C.) said Friday.

A two-thirds majority in the House and Senate is needed to amend the Constitution, so the amendment appears doomed. The Senate expects to hold its own vote on the amendment by midweek.

Democratic and Republican aides said that once it is clear the votes are not there to send a constitutional amendment to the states, Reid will move on the Senate plan.

The president said Friday that the Reid-McConnell effort needs to address “the underlying problems” of the nation’s $14.3 trillion — and rising — debt.

“If Washington operates as usual and can’t get anything done, let’s at least avert Armageddon,” Obama said.