With the Treasury reaffirming a deadline that is barely four weeks away, Congress headed home Friday for a holiday weekend having accomplished nothing the past nine days except partisan posturing in the effort to shore up the nation’s finances.

Congressional negotiators and President Obama ended up no closer to a deal that could carve out more than $2 trillion in savings and allow the Treasury to continue borrowing money to finance the federal government. Any chance for some extra breathing room dissipated Friday when aides to Treasury Secretary Timothy F. Geithner confirmed that Aug. 2 remained the deadline for raising the $14.3 trillion debt limit. On that date the United States will begin to default on its obligations without additional borrowing authority, setting in motion the “catastrophic economic and market consequences of a default crisis,” according to a Treasury statement.

Yet an influential bloc of optimists inside the White House and on Capitol Hill, on both sides of the aisle, are confident a deal can be reached. To this group the past week’s showmanship was a necessary evil, with Obama talking to his liberal base and Senate Minority Leader Mitch McConnell (R-Ky.) shoring up his conservative flank. Meanwhile, the talks led by Vice President Biden, which House Majority Leader Eric Cantor (R-Va.) brought to a standstill last week by walking out amid Democratic demands for increased taxes, were always meant to be turned over to Obama and the top congressional leaders, the optimists say.

The Biden talks produced a range of options for Obama, House Speaker John A. Boehner (R-Ohio) and Senate Majority Leader Harry M. Reid (D-Nev.) as they try to assemble $2 trillion in savings to meet Boehner’s demands and provide for a increase in the debt ceiling that would last beyond the 2012 election.

“The way these things work, there’s a lot of two steps forward, one step back,” said Sen. Charles E. Schumer (D-N.Y.).

Administration officials see July 22 as the deadline to reach a deal and leave enough time to formally draft the package and move it through both houses of Congress. But some lawmakers say it’s already too late to draft legislation of the size and complexity under discussion.

This group believes there will be a short-term extension of the debt ceiling — or a brief period of default — as the final details are approved through a summer session that lasts deep into August.

Outside the talks are rank-and-file pessimists. Those lawmakers, with only a vague knowledge of the details, have been whipsawed since the Biden talks broke down. On Wednesday, Obama accused GOP leaders of having less maturity than his 12-year-old and 10-year-old daughters, then one Senate Republican countered by telling Obama to “take a Valium.”

People in both parties said the Biden-led talks had been remarkably productive, with full discussions about every major budget category, including Democratic proposals to raise revenue. The group had tentatively agreed on more than $1 trillion in spending cuts over the next decade and were “within shouting distance” of additional savings when Republicans “kicked it up to the leader level,” said a Democrat familiar to the talks, who like others interviewed for this story spoke on the condition of anonymity to discuss the closed-door negotiations.

The pot of consensus savings includes about $200 billion from Medicare and Medicaid and another $200 billion from other direct-payment programs, such as farm subsidies and federal employee pensions.

 Major disagreements remain. For example, House Republicans want as much as $1.7 trillion in agency savings through 2021. And they have so far refused to agree to Democratic demands for a “firewall” between defense and non-defense spending that would force lawmakers to distribute the pain broadly across the government, hitting the military as well as domestic agencies. A Democratic official close to the talks called the firewall “one of the more contentious issues.”

On taxes, the White House has offered what administration officials describe as a “menu of options” that could generate more than $400 billion over the next decade. Proposals include limiting the value of deductions for the nation’s wealthiest families, changing the way inventory is valued for tax purposes and wiping out special perks for owners of corporate jets, the oil and gas industry, and hedge fund managers.

Republicans continue to oppose any tax increases.

“The disagreement remains over hundreds of billions of dollars in tax hikes that Democrats are trying to impose on individuals, small businesses, and employers,” Brad Dayspring, Cantor’s spokesman, said Friday in a statement.

Democrats say they are insisting revenue be part of the package for two reasons: First, it’s hard to significantly rein in debt solely by cutting spending. Second, Democrats – whose votes will be needed to pass any package – are demanding “a balanced approach.”

Democrats have not insisted that taxes be a large part of the deal, however. “We don’t have an imaginary number that it has to be,” said a Democratic official close to the talks. “The entire exercise is to get 218 votes in the House.”

People in both parties say they can see a number of paths forward. Several senators and senior aides said that there is a likelihood that the deal would be drafted in a way that would include some immediate savings — particularly in the programs from mandatory spending — but that other savings could come from a complex process that is unlikely to end with final passage of this legislation.

“You’re dealing with a very complicated piece of business here,” said Senate Budget Committee Chairman Kent Conrad (D-N.D.). “That’s not going to be concluded by August 2nd.”

Staff writers Rosalind S. Helderman and Felicia Sonmez contributed to this report.