Democratic leaders vowed to kill the House measure in the Senate without a vote, urging Speaker John Boehner to head back to the negotiating table with President Barack Obama.
The solution to avoid the fiscal cliff's automatic tax hikes and spending cuts is still not clear. Senate Majority Leader Harry Reid, D-Nev., said lawmakers planned to return to the Capitol after the holiday to try again.
"We don't need a vehicle. We need an agreement," said Sen. Charles Schumer, D-N.Y. "The president and Boehner have to negotiate this, OK?"
Boehner has accused the president and his fellow Democrats of doing nothing to prevent the so-called cliff - wide-reaching tax hikes and spending cuts that is expected to hit in early January - unless lawmakers head them off. But at the same time, he told reporters he would continue trying to strike a deal with the president.
"The president and I in our respective roles have a responsibility to work together to get them resolved. I expect that we'll continue to work together," said Boehner, R-Ohio.
The bill dubbed "Plan B" by Boehner aims at upping the year-end pressure on Capitol Hill Democrats and Mr. Obama. House Majority Leader Eric Cantor, R-Va., said the GOP has enough votes to pass it. But in a brief statement late Thursday, Boehner recoiled, saying the bill "did not have sufficient support from our members to pass."
The White House has said the president would veto Plan B. Spokesman Jay Carney called the GOP move a fruitless step.
Republicans have told senior administration officials that the reason why Boehner decided to offer his Plan B was because he couldn't get enough GOP support for the proposal he made to Mr. Obama over the weekend.
GOP officials said while Cantor supports Boehner's efforts to reach a deal with the president, other members of the leadership team recoiled at details of Boehner's latest plan at a Monday night meeting. His plan includes $1 trillion in higher taxes and a breakthrough concession on higher tax rates for those earning more than $1 million.
On Wednesday, Mr. Obama dismissed the GOP bill, telling reporters that he and Boehner were just a few hundred billion dollars apart on a 10-year, $2 trillion-plus deficit-cutting pact
At the White House on Wednesday, Mr. Obama said Republicans should "peel off the partisan war paint" and take the deal he's offering. He added that he had won re-election with a call for higher taxes on the wealthy.
On paper, the two sides are relatively close to an agreement on major issues, each having offered concessions in an intensive round of talks that began late last week.
The talks have stalled, though Mr. Obama and Boehner have each made concessions that would seem to bring them to the brink of agreement. Obama is now seeking $1.2 trillion in higher tax revenue, down from the $1.6 trillion he initially sought. He also has softened his demand for higher tax rates on household incomes so they would apply to incomes over $400,000 instead of the $250,000 he cited during his successful campaign for a new term.
Mr. Obama also has offered more than $800 billion in spending cuts over a decade, half of it from Medicare and Medicaid, $200 million from farm and other benefit programs, $100 billion from defense and $100 billion from a broad swath of government accounts ranging from parks to transportation to education.
In a key concession to Republicans, the president also has agreed to slow the rise in cost-of-living increases in Social Security and other benefit programs, at a savings estimated at about $130 billion over a decade.
By contrast, Boehner's most recent offer allowed for $1 trillion in higher taxes over a decade, with higher rates for annual incomes over $1 million. His latest offer seeks about $1 trillion in spending cuts.
The Associated Press contributed to this report.
