Vice President Joseph Biden wants a debt limit package ready for Congressional leaders by the Senate’s July Fourth recess, he said Tuesday after another round of bipartisan negotiations.

“We are making real progress,” he said after the meeting in the Capitol. “I’m convinced — it ain’t over until it’s over — but I’m convinced we can come up with an agreement that gets the debt limit passed and makes some real serious down payment on the commitment to 4 trillion bucks over the next 10 to 12 years.”

Biden is leading a group of lawmakers in negotiating an agreement to raise the debt ceiling and reduce the deficit. The group started meeting in early May, and the U.S. Treasury says the government risks going into default if the ceiling isn’t increased by Aug. 2.

“We all have agreed that to just keep this thing going around the clock, basically,” in order to come to an agreement, Biden said.

“I think we’re going to be in a position, hopefully, that by the end of the month, by the Fourth of July recess, we have something to take to the leaders and actually get down to the implementation piece of what kind of legislation is needed. That’s the goal,” he added.

Biden said that he expects the debt limit package to go “well beyond a trillion” dollars in cuts.

Still, Biden and the group’s members ­­— including House Budget ranking member Chris Van Hollen (D-Md.), Assistant Leader James Clyburn (D-S.C.) and Senate Minority Whip Jon Kyl (R-Ariz.) — gave scant details when exiting the meeting, which lasted more than two hours. They are set to resume talks at 9 a.m. Wednesday.

The group was scheduled to discuss the budgeting process Tuesday, and the talks primarily focused on discretionary spending, according to Van Hollen.

“It went well. We’re getting into some of the tougher issues,” the Maryland Democrat said. “The fact is we’re still all friends and we’re talking around the table.”

Van Hollen said that although they shouldn’t set an artificial deadline for finishing the deal, a final package should be completed well before the country would default on its debt obligations.

It’s unclear how close Republican and Democratic lawmakers are to agreeing on a final package. Senate Democratic leaders held a news conference earlier in the day to continue hammering their message that Medicare should not be a part of a final agreement.

Senate Majority Leader Harry Reid (D-Nev.) also declined to discuss his preferences on what should be included in a deal, saying it would be inappropriate while Senate Appropriations Chairman Daniel Inouye (D-Hawaii) is engaged in negotiations. Senate Finance Chairman Max Baucus (D-Mont.) and House Majority Leader Eric Cantor (R-Va.) are the other Congressional members of the group.

Reid said he would “leave the process as it should be” and that he would weigh in if the group asks for his opinion.

House Minority Whip Steny Hoyer told reporters earlier in the day that he wanted the deal done by the end of the month.

“Speaker Boehner indicated that we need to do this by the end of this month. I believe he’s absolutely right,” the Maryland Democrat said. “We ought not to wait until July, we ought to act within the next 10 days to make sure the markets know that America’s going to pay its bills and that we’ll lift the ceiling to allow that to happen.”

Hoyer reiterated his call to raise the debt limit by enough to carry the nation through the November 2012 elections in order to “depoliticize” the debate, and he said that new revenues should be part of any deficit reduction deal. If a bipartisan deal is ultimately reached, Hoyer predicted that “well over half of our Caucus” would vote in favor of raising the debt ceiling.

Jessica Brady contributed to this story.