Washington (CNN) -- Vice President Joe Biden held another round of high-stakes budget talks with top congressional negotiators Tuesday -- one day after House Republicans demanded trillions of dollars in new spending cuts as their price for agreeing to raise the federal debt ceiling and avoid a potential economic catastrophe.
Biden and Treasury Secretary Tim Geithner met with, among others, House Majority Leader Eric Cantor, R-Virginia; Senate Appropriations Committee Chairman Daniel Inouye, D-Hawaii; Senate Finance Committee Chairman Max Baucus, D-Montana; and Sen. Jon Kyl, R-Arizona, a top member of the Finance Committee.
South Carolina Rep. James Clyburn and Maryland Rep. Chris Van Hollen -- both part of the House Democratic leadership -- also attended.
"I remain optimistic. We had a really good discussion," Biden said he left the meeting. "We have set an agenda for the next meeting and we're making real progress. Whether we get to the finish line with this group is another question. But everybody is being straight, cordial, all the facts are being laid on the table."
The vice president said members of the group "agree on a lot" and that he would give reporters specifics "when it's over."
Baucus and Clyburn also described the talks in favorable terms as they left the meeting, with Clyburn flashing a "thumbs up" sign.
President Barack Obama has said he hopes the talks, which began last week, will lead to an agreement by June. The president is scheduled to discuss the issue during a separate meeting with Senate Democrats on Wednesday and Senate Republicans on Thursday.
The goal: agreeing on a deficit reduction strategy that is at least minimally acceptable to both Democratic and Republican base voters.
"We believe ... that common ground can be found between Republicans and Democrats in these negotiations," White House press secretary Jay Carney told reporters before the meeting. "We continue to be optimistic."
Geithner has said the government is extremely close to reaching the current $14.294 trillion debt ceiling. The secretary indicated in a recent letter to Congress he can keep the country out of default until August 2.
Economic analysts warn that a failure to reach an agreement in time to avoid a national default could have devastating consequences. Among other things, Americans could be faced with skyrocketing interest rates and a plummeting dollar.
Republicans have repeatedly said they will not agree to any tax hikes as part of a deal with Democrats. House Speaker John Boehner, R-Ohio, raised the stakes Monday night during an address to the New York Economic Club, declaring that the overall size of any spending cuts has to exceed the magnitude of the ceiling increase.
"Without significant spending cuts and the way we spend Americans' money, there will be no debt limit increase," Boehner declared. "The cuts should be greater than the accompanying increase in debt authority the president is given."
Boehner stressed that the spending cuts should be in the "trillions, not just billions," and that simply agreeing to spending and deficit goals for the future won't be sufficient.
"They should be actual cuts and real reforms to these programs, not broad deficit or debt targets that punt the tough questions to the future," the speaker declared. "With the exception of tax hikes -- which in my opinion will destroy jobs -- everything is on the table."
The speaker acknowledged Monday that a default would be dangerous, but said it would be even worse for the country to continue on its current fiscal course.
"It's true that allowing America to default would be irresponsible," he said. "But it would be more irresponsible to raise the debt ceiling without simultaneously taking dramatic steps to reduce spending and reform the budget process."
Boehner didn't specify the exact size of the GOP's desired cuts. Congressional sources, however, have told CNN they believe the Republican-controlled House could vote to increase the debt ceiling by about $2 trillion.
In contrast, the entire annual federal budget is roughly $3.5 trillion.
Boehner refused to specify the period of time over which any spending cuts could be implemented -- a critical factor in any agreement. Kyl has suggested $6 trillion in cuts over 10 years, but there has been no public agreement on those numbers within the Republican caucus.
It is also unclear exactly how much Congress is willing to increase the debt ceiling. One GOP aide told CNN on Tuesday it is hard to imagine not agreeing to a debt limit increase that lasts at least through the end of the 2012 election cycle.
Any vote to increase the debt -- even if it's tied to major spending cuts -- is "toxic" to Republicans, the aide said.
Democrats have called the Republican position extreme and blasted the GOP for refusing to be more flexible in the talks.
"Maximalist positions do not produce compromise," Carney said Tuesday.
Most people "say that the best way to reduce the debt is to tax the millionaires and billionaires in our country," Rep. Jan Schakowsky, D-Illinois, told CNN Monday night. "We can solve our problem. ... (Democrats are) willing to make cuts. We are willing to do that. But you have to put revenue on the table as well."
While the political stakes are high in the debt ceiling negotiations, those talks are only one facet of Washington's current fiscal challenge. Democrats and Republicans agreed last month to a package of $38.5 billion in spending cuts covering the remainder of the current fiscal year. They are also trying to find common ground on the budget for fiscal year 2012, which starts October 1.
The current GOP proposal for the next fiscal year, drafted by House Budget Committee Chairman Paul Ryan, R-Wisconsin, would cut federal deficits by roughly $4.4 trillion over the next decade. It would overhaul key portions of the tax code, dropping the top rate for individuals and businesses to 25% while eliminating a number of loopholes.
The most contentious parts of Ryan's blueprint revolve around its proposed changes to Medicare and Medicaid. Medicare, a major contributor to growing federal deficits, would be overhauled starting in 2022. The government would no longer directly pay bills for senior citizens in the program. Instead, recipients would choose a plan from a list of private providers, which the federal government would subsidize.
Individuals currently 55 or older would not be affected by the changes.
Medicaid, which provides health care for the disabled and the poor, would be transformed into a series of block grants to the states. Republicans believe that state governments would spend the money more efficiently and would benefit from increased flexibility. Democrats warn that the move would shred the health care security provided to the most vulnerable Americans in recent generations.
GOP leaders have denied reports that they are already backing away from their Medicare plan. Democrats, who control both the Senate and the White House, have repeatedly said the idea has no chance of being enacted.
Obama's plan, in contrast to Ryan's, aims to cut deficits by a combined $4 trillion over the next 12 years without significantly changing any of the major entitlements. Among other things, the president wants to repeal the Bush-era tax cuts on families making more than $250,000, something vehemently opposed by Republicans.
The president has also called for the creation of a "debt fail-safe" trigger that would impose automatic across-the-board spending cuts and tax changes in coming years if annual deficits are on track to exceed 2.8% of the nation's gross domestic product.
Administration officials claim that by building on or adjusting the health care reform bill passed last year, $480 billion would be saved by 2023, followed by an additional $1 trillion in the following decade. Obama has also proposed tightly constraining the growth in Medicare costs starting in 2018.
CNN's Dana Bash and Jeanne Sahadi contributed to this report