A grand bargain in debt talks between Democrats and Republicans remains elusive amid Washington’s partisan sniping, as the deadline to avoid a government default draws closer.

President Obama said at the start of a White House meeting with congressional leaders Sunday night that “we need to” come together on a budget deal within 10 days. But the session broke up after 80 minutes without any progress. About the only tangible agreement was to meet again today.

Sources familiar with the meeting said Obama again made a pitch for a “big” budget agreement, and the Republicans again told him the votes weren’t there—pretty much a replay of the last meeting, except that the leaders were tieless for the Sunday session.

The president will hold a news conference, before Monday’s talks get underway, at 11 a.m.—which suggests he is determined to build public pressure on the Republicans to make concessions. But Obama must walk a fine line between making the GOP lawmakers appear recalcitrant and angering them through harsh rhetoric.

The odds of anything other than a short-term fix grew considerably longer over the weekend. Indeed, the Republicans seemed more deeply dug into their position than just days earlier.

The president of the United States and the speaker of the House may think it’s a grand idea to compromise on a sweeping $4 trillion plan to slash the deficit and raise the debt ceiling. But neither had the ability to bring the parties along.

John Boehner came under such fierce pressure from the right that on Saturday he had to abandon the $4 trillion plan, which would have included significant tax increases, and backpedal toward a $2 trillion fix, which is still going to be difficult to achieve before the Aug. 2 deadline for raising the debt ceiling.

“Boehner wanted to be a hero, but he got his legs cut out from under him,” says a senior Democrat close to the talks. “Their caucus went into a tizzy.”

The fast-moving events revealed a real split between Boehner and his House majority leader, Eric Cantor, who has taken a harder line against raising revenue and argued that a $2 trillion target was more realistic.

But Obama also faced a revolt on the Democratic side, led by Nancy Pelosi. The House minority leader and other lawmakers openly rebelled against the Medicare cutbacks that the administration put on the table as a kind of tradeoff for the revenue hikes the GOP was being asked to swallow. The Pelosi wing views Medicare as the most effective weapon with which to club the Republicans in next year’s elections, since the GOP has embraced turning it into a voucher program.

Some politicians involved in the process say privately they now expect to run out of time and to be forced to cobble together a short-term deal that simply kicks the can into 2012—precisely what Obama, Boehner, and other leaders have said they don’t want to do. But the reality is, their options are dwindling.

It isn’t for lack of trying. Even after the budget talks led by Vice President Biden collapsed, Obama and GOP leaders scrambled behind the scenes, recognizing that they would pay a huge price—and shatter the fragile economic recovery—by failing to boost the debt ceiling before the deadline.

The problem comes down to political math. Even if Boehner wants to swallow the closing of some tax loopholes to give Democrats political cover for painful spending cuts, he would lose somewhere between 50 and 80 GOP votes for such a deal. So many Republicans, with Tea Party backing, ran on a no-new-taxes pledge last year that they are extremely reluctant to embrace anything that looks or smells like a tax. Cantor has said again and again that the votes aren’t there for any package that boosts revenue.

The result is that Boehner will need a significant chunk of Democrats to join him, but many of Pelosi’s troops are balking as well. Anything other than token cuts in Medicare is anathema to a party that has been running on the LBJ-era program for half a century.

And therein lies the dilemma. Without health-care cutbacks and tax increases, it is awfully hard to reduce the deficit by more than $1 trillion, which is the level of spending reductions hammered out in the Biden talks. And with those elements, support declines dramatically on the right and left.

The GOP’s congressional wing is feeling the heat from another direction: the party’s presidential candidates. Michele Bachmann and Tim Pawlenty are among those urging against a compromise package. In her first television ad, Bachmann, a third-term House member, says flatly: “I will not vote to increase the debt ceiling.”

Senate Democrats, meanwhile, have drafted a $4 trillion debt-reduction plan that doesn’t touch Medicare or Social Security but relies on $2 trillion in new taxes, according to The Washington Post. The new levies would fall primarily on families earning more than $1 million a year. With GOP leaders balking at even relatively modest measures to close tax loopholes for ethanol, oil, and farm products, the Senate Democratic blueprint seems like a nonstarter.

No one knows what will happen if the impasse drags on, but Senate Minority Leader Mitch McConnell told Fox News he has a “contingency plan” in his back pocket—undoubtedly code for a short-term compromise to get the country past the default date.

The politicians were back to their usual talking points on the morning talk shows, although White House Chief of Staff Bill Daley did acknowledge that Washington’s “dysfunction” is hurting business. “I do firmly believe that one of the wet blankets on this economy—and on companies, on the system right now—is a question as to whether or not our political system, whether the leaders can get together, whether they can solve big problems, or are they just going to kick the can?” he said on ABC’s This Week.

On Fox News on Sunday, GOP Sen. Jim DeMint minimized the importance of the deadline and accused the president of “gaming” Republicans.

“He’s playing Chicken Little here," DeMint said. "The fact is, we will pay our debts if it’s the last dollar we have. There are enough assets in Social Security and Medicare to pay the benefits of those programs for several years. Other programs can be funded from tax revenue.”

The spending cuts–vs.–tax hikes argument played out on CNN’s State of the Union. Rep. Kevin McCarthy, the House majority whip, said the talks hit a snag because the Democrats are insisting on boosting revenue (in addition to offering $1 trillion in spending cuts). “In principle, no, we’re not going to go there…I’ve never found one tax increase that created a job,” McCarthy said.

Democratic Rep. Chris Van Hollen countered: “They walked out, because they did not want to take away the tax breaks for corporate jets for the purpose of deficit reduction, oil and gas companies, and folks at the very top.”

The unyielding rhetoric makes clear that, for the moment at least, both sides are firmly dug in.