President Barack Obama had urged the House to pass the bill "without delay".
But Eric Cantor, the House majority leader, emerged from a two-hour party meeting to voice opposition, saying: "I do not support the Bill."
Analysts say Mr Cantor's opposition would destroy the chances for a fast House passage of the legislation without changes, potentially pushing the country over the "fiscal cliff" after all, despite months of effort. Many economists believe the fiscal cliff's full effect would drive the world's largest economy back into recession and have a global impact.
Republicans said they might add more spending cuts to the bill, which contains $600 billion in tax increases but only around $1bn in spending cuts.
It remained unclear when the House will consider the Senate bill, which was approved with an overwhelming 89-8 vote well after midnight on New Year's Day.
Republicans could face a backlash if they quash the deal. Income tax rates technically rose back to 1990s levels for all Americans at midnight, and across-the-board spending cuts on defence and domestic programs are due to kick in on Wednesday.
Economists say the $600bn combination of tax cuts and spending cuts could push the economy into recession, and public opinion polls show Republicans would shoulder the blame.
The US Senate voted for a new deal in a rare New Year's Day session, which raised some taxes on the wealthy while making permanent low tax rates on the middle class that have been in place for a decade.
But the agreement came too late for Congress to meet its own deadline of New Year's Eve for passing laws to halt tax hikes and spending cuts which strictly speaking came into force yesterday.
Because of the New Year's Day holiday, there was no real world impact and Congress still had time to draw up legislation, approve it and backdate it to avoid the harsh fiscal measures.
But that needed the backing of the House where many of the Republicans who control the chamber complain Mr Obama has shown little interest in cutting government spending and is too concerned with raising taxes.
Mr Obama called for the House to act quickly and follow the Senate's lead. He said: "While neither Democrats nor Republicans got everything they wanted, this agreement is the right thing to do for our country and the House should pass it without delay. There's more work to do to reduce our deficits and I'm willing to do it. But tonight's agreement ensures that, going forward, we will continue to reduce the deficit through a combination of new spending cuts and new revenues from the wealthiest Americans."
But if politicians cannot pass legislation in the coming days, markets are likely to turn sour and the US economy could stall again.
Majority Leader Harry Reid, a Democrat, told the Senate floor: "If we do nothing, the threat of a recession is very real. Passing this agreement does not mean negotiations halt, far from it."
A new, informal deadline for Congress to legislate is set for today when the current body expires and is replaced by a new Congress chosen at last November's election.
The Senate bill was worked out after long negotiations on New Year's Eve between Vice-President Joe Biden and Senate Republican Minority Leader Mitch McConnell.