House Speaker John Boehner dashed cold water Thursday on claims that the two parties had agreed on how much to cut from the rest of this year's budget, saying Republicans would "continue to fight" for an earlier budget proposal Democrats adamantly oppose.

Democrats had earlier claimed negotiators were working on a proposal for around $33 billion in spending cuts. Vice President Biden reported "good progress" on budget talks, and Senate Majority Leader Harry Reid, D-Nev., went so far as to claim the parties had "agreed upon a number."

But while that target is indeed on the table, Boehner said Republicans have not agreed to it.

"There's no agreement on numbers, and nothing will be agreed to until everything is agreed to," Boehner said. "We're going to continue to fight for the largest spending cuts that we can get."

He said that means Republicans will keep fighting for the GOP-backed budget that passed the House last month and stalled in the Senate. That budget proposal would cut $61 billion from last year's levels, and contains myriad cuts to organizations like Planned Parenthood which Democrats decry as mere political score-settling.

Democrats have hammered the message that Republicans are reticent to come down from the $61 billion mark because they're under pressure from Tea Partiers. "I'm sure it's not easy trying to negotiate with the Tea Party screaming in his right ear," Reid said in a statement Thursday.

Both parties are casting the other as "extreme," amounting to a public-relations battle that has made the negotiations themselves all the more fragile.

The tentative split-the-difference plan would end up where GOP leaders started last month as they tried to fulfill a campaign pledge to return spending for agencies' daily operations to levels in place before President Obama took office. That calculation takes into account the fact that the current budget year, which began Oct. 1, is about half over.

The $33 billion figure is well below the cuts the House passed last month, but it still represents significant movement by Senate Democrats and the administration after originally backing a freeze at current rates.

"There's no reason why, with all that's going on in the world and with the state of the economy, that we can't avoid a government shutdown," Biden told reporters after a meeting in the Capitol with Senate Democratic leaders.

Some Tea Party-backed GOP lawmakers want the original $61 billion. With a Tea Party rally set for Thursday on Capitol Hill, it's unclear how many of the 87 freshmen Republicans elected last fall could live with a smaller compromise.

Both sides said the figure under consideration is tentative at best and depends on the outcome of numerous policy stands written into the bill.

Some conservatives appear insistent on the full range of spending cuts, but others recognize that compromise is required to win Obama's signature and support from Democrats who control the Senate.

Far bigger fights are ahead on a longer-term GOP budget plan that takes a more comprehensive approach to the budget woes. Also looming is a must-pass bill to allow the government to borrow more money to meet its commitments. Republicans hope to use that measure to force further spending cuts on the president.

"I don't believe that shutting down government is a solution to the problem. Republicans and Democrats need to work out a compromise," said Rep. Charles Bass, R-N.H. "Let's get this over with and get on to the budget."

But Rep. Mike Pence, R-Ind., who earlier warned that "It's time to pick a fight," wants party leaders to hang tough.

The legislation would bankroll the day-to-day operating budgets of federal agencies -- including the wars in Iraq and Afghanistan -- through Sept. 30, the end of the current budget year.

The Associated Press contributed to this report.