After six weeks of talks with congressional leaders aimed at restraining the spiraling national debt, Vice President Biden emerged Thursday with a blunt message: Now, the hard part begins.

Next week, Biden said, negotiators from the White House and Capitol Hill will begin working “around the clock” to bridge the yawning philosophical divide between the two parties, as Democrats press for fresh revenue and Republicans push for significant cuts to federal health programs as part of the debt-reduction package.

“Now we’re getting down to the real hard stuff: I’ll trade you my bicycle for your golf clubs,” Biden told reporters, adding that he remains optimistic that the talks will produce a tentative agreement before the July 4 break.

“Everyone wants an agreement,” Biden said. “There’s no principal in that room that doesn’t want to get an agreement that bends the curve on the long-term debt, and that is sufficiently realistic to get us to $4 trillion over a decade or so in terms of [debt] reduction.”

Biden spoke after emerging from the latest two-hour bargaining session at the U.S. Capitol with top administration officials and six lawmakers from the House and Senate. The group, which began meeting May 5, is rushing to assemble a debt-reduction package to ease passage of an increase in the legal limit on government borrowing, which is set at $14.3 trillion.

The national debt is already bumping up against that limit. Unless Congress acts, the government risks defaulting on its obligations after Aug. 2. Lawmakers in both parties have said they will not vote to raise the debt limit without a plan to dramatically restrain future borrowing.

So far, negotiators have plowed through the major categories of the federal budget, looking for areas of common ground. Democrats and Republicans are interested in cutting farm subsidies, for example, and both sides want to sharply slow the growth of spending on government agencies.

“We’ve gone through all the discrete elements of the budget and we’ve said, ‘Okay, if we have an agreement on everything, we agree on this piece. . . . We can save $10 here,’ ” Biden said. But, he added, an agreement will not be sealed until the parties address larger differences “that are going to have to be bridged.” And that, he said, “won’t occur till the end.”

Biden said that Democrats will insist that any deal include new revenue, and that the White House will continue pushing for cuts in military spending. So far, Republicans have resisted defense cuts and have refused to consider any increases in taxes.

Biden said Democrats are open to reaping savings from federal health programs, including Medicare, which is projected to be the biggest driver of future spending.

“I don’t mean major Medicare reform, but just changes in health policy,” he said.

Possibilities include demanding larger discounts from drug companies and lower payment rates for health-care providers. The group has also discussed charging well-off beneficiaries more for Medicare, according to people familiar with the talks, an idea endorsed by House Republicans and President Obama’s fiscal commission.

The two sides appear to agree on the magnitude of the package: at least $2 trillion in savings — not including interest on the debt — to pair with an increase in the debt limit of a similar size, according to people familiar with the talks. The government needs at least $2 trillion in fresh borrowing authority to pay the bills through the 2012 election, and House Speaker John A. Boehner (R-Ohio) has said House Republicans will insist that the increase be met with dollar-for-dollar savings.

House Majority Leader Eric Cantor (R-Va.) agreed that the group is making progress.

“We are continuing to try to push forward to the kind of goals that have been set out by the speaker and others,” he told reporters.

Though any debt-reduction plan is likely to be politically painful, Biden said the group is determined to produce a package that can pass the Republican-controlled House and the Democratic-controlled Senate and convince U.S. investors and world financial markets that Washington can muster the political will to avoid the kind of debt crisis that has sparked riots in Greece.

“What is the schtick out there? That this is a dysfunctional place. Can’t get anything done,” Biden said of Washington. “The single most important thing to do for the markets is convince them that no, no, no, that’s not true. We can handle significant decisions and make them.”

Staff writer Felicia Sonmez contributed to this report.