Tea Party adherents demand even larger cuts, which Democrats warn could harm the economic recovery US congressional leaders continue to labour on a deal that could cut a record $33bn (£20.6bn) from the government budget by October.
But Republican House Speaker John Boehner warned on Thursday that a final agreement had not been reached.
Members of the anti-big government Tea Party movement held a rally in Washington DC to demand greater cuts.
Without a budget agreement of some kind, much of the US government will shut down on Friday 8 April.
Mr Boehner and Democratic negotiators including Vice-President Joe Biden and Senate Majority Leader Harry Reid have reportedly agreed tentatively to the $33bn figure, although Mr Boehner said on Thursday the figure was not final.
“We are going to fight for all of the spending cuts that we can get,” Mr Boehner, of Ohio, told reporters.
“We control one half of one third of the government here in Washington. We can’t impose our will on another body. We can’t impose our will on the Senate. All we can do is to fight for all of the spending cuts that we can get an agreement to.”
Many analysts say $33bn in cuts would be a victory for Republicans, even though the party’s House caucus had earlier succeeded in passing a bill that trimmed $61bn from the budget by October while stripping funding from President Barack Obama’s signature healthcare reform effort and other Democratic priorities.
That bill failed in the Democratic-controlled Senate, and Mr Obama had vowed to veto it anyway.
A $33bn cut – the single largest budget cut in US history – would severely hamper many government programmes and Democrats have warned large cuts could endanger the fragile economic recovery.
But the cuts would do little to curb the current $1.4 trillion budget deficit, and Republicans remain steadfastly opposed to any tax increases.
The government has been operating without a long-term budget since the start of the fiscal year on 1 October, subsisting instead on a series of temporary measures, the most recent of which was passed two weeks ago and expires on 8 April.
That and a previous measure had already slashed $10bn from spending by 8 April, mostly on programmes Mr Obama had already identified for cuts.
Without an agreement in Congress, many government services will be shut down for lack of funding, an outcome both parties say they hope to avoid.
“There are hurdles to clear before we get from here to a deal but… progress is being made,” White House spokesman Jay Carney told reporters.
Meanwhile, a group of Tea Partiers who rallied on Capitol Hill on Thursday demanded even steeper cuts and urged Republican Congressional leaders not to compromise with Democrats.
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