After weeks of arguing, Republicans and Democrats on Capitol Hill began negotiations Wednesday on a possible budget agreement that would slash federal spending by as much as $33 billion and avert a government shutdown.

“We’re all working off the same number now,” Vice President Biden told reporters after meeting with Senate Democratic leaders at the Capitol on Wednesday evening. “Obviously, there’s a difference in the composition of that number — what’s included, what’s not included. It’s going to be a thorough negotiation.”

If approved, the deal would be the largest single-year budget cut in U.S. history.

Lawmakers in both parties are eager to reach such a compromise, which would fund the government through the end of the fiscal year, in September, and end a series of stopgap spending resolutions that have kept Washington operating a few weeks at a time since last fall. The current short-term measure will expire April 8, and congressional leaders have said they don’t want to pass another one.

The two sides have already agreed on $10 billion in cuts; now, the House and Senate appropriations committees are searching for an additional $23 billion to extract from the budget, according to lawmakers and aides from both parties.

“We’re going to try to find some common ground,” House Appropriations Chairman Harold Rogers (R-Ky.) told reporters. “It’s going to take some time. . . . [But] the leadership has said for us to get started.”

Congressional leaders cautioned that no final deal has been reached. The talks could break down over disputes about how much to cut and from where.

“There have been discussion for weeks, and those discussions are continuing,” said Michael Steel, a spokesman for House Speaker John A. Boehner (R-Ohio). “There’s no agreement, and nothing will be agreed to until everything is agreed to.”

Boehner on Thursday dismissed reports that agreement on a number had been reached.

“You’ve heard a lot of talk over the last 24 hours,” Boehner said at a Capitol news conference. “There is no agreement on numbers. Nothing will be agreed to until everything is agreed to.”

He emphasized that House Republicans will continue to push for greater cuts, although he noted that their leverage remains limited because they retain the majority in only one chamber of Congress.

“We control one-half of one-third of the government here, but we’re going to continue to fight for largest spending cuts that we can get and to keep the government open and funded through the balance of this fiscal year,” Boehner said.

Some conservative House Republicans — led by freshmen who came to Washington on a promise to shrink the government — have said they would vote against any proposal that falls short of the $61 billion in reductions the House approved on a party-line vote last month. Senate Democrats immediately rejected that measure.

Spending cuts are not the only issue up for negotiation. As part of their initial budget package, Republicans included unrelated amendments, called “riders,” that would impose restrictions on federal agencies. Democrats have objected to many of them, including one that would prohibit federal funding to Planned Parenthood and another that would weaken the Environmental Protection Agency’s power to regulate carbon emissions.

Some Republicans have suggested that in exchange for giving up some of the spending cuts they want, they will pressure Democrats to accept at least some of these provisions.

On Wednesday, the vice president indicated that such an agreement was at least a possibility, although he did not give details or say which riders Democrats might be willing to accept.

The progress in the talks came on the eve of a planned rally Thursday by tea party activists on the Capitol lawn, where leaders of the conservative movement are expected to call for no compromise with President Obama and the Democratic-controlled Senate.

House Republicans fear that many rank-and-file GOP lawmakers could view a deal with Democrats as a retreat. Conservative Republicans warned that any agreement would require their support in large numbers, indicating that Boehner would not back a spending plan that evenly divided their 241 members.

The speaker and his leadership team have sought to win the support of their colleagues by promoting a budget compromise as only the first part of a “three bites at the apple” strategy to reduce the size of government. The idea, they say, is to prune as much money as they can from this year’s budget, then move on to bigger fiscal issues: the debate over whether to raise the federal debt limit, expected later this spring, and the negotiations over the 2012 budget. Each of those battles will provide GOP leaders the opportunity to demand even larger cuts.

In the event that Boehner loses the support of two dozen or more of his GOP colleagues, he could turn to moderate Democrats for support. Republican leaders met privately with a group of Democrats but stressed Wednesday that those talks focused mostly on longer-term deficit reduction and entitlement reform — issues awaiting congressional action once they get past the current spending fight.

Democrats are also split over how far they are willing to go in compromising with Republicans. Senate Majority Leader Harry M. Reid (Nev.) faces a protest from liberal Senate Democrats, including advocates for abortion rights and environmental causes who object to the idea that the Planned Parenthood and EPA riders might be part of a deal.

Reid brought Biden to the Capitol to reassure Democratic leaders that the White House would not undercut them in talks with Boehner.

“The main reason to be here today is to make sure that Democrats in the Senate and the president and I are on the same page,” Biden said. “We’re on the same page.”

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