House Speaker John Boehner has reportedly presented fellow Republicans two options to overcome the impasse on the Senate-approved bill to avert the so-called fiscal cliff.
Citing a senior House Republican aide, The Wall Street Journal reports that Mr Boehner briefed his party members on the proposals during a closed-door meeting Tuesday afternoon.
The aide said one option included adding a spending cut package to the Senate version of the bill, which was approved with an overwhelming 89-8 vote well after midnight on New Year's Day.
Republicans are said to be conducting a quick count to determine if the amendment to the bill would garner the votes needed to pass it through the House before it is sent back to the Senate for approval.
The second option would be to leave the bill alone and allow the House to vote it up or down as it stands.
But the fallback option does not appear to bode well for the bill's passing, with some Republican leaders expressing concern over the legislation in its current form.
The number two ranking Republican in the House, Eric Cantor, voiced opposition to the Senate-approved bill earlier Tuesday.
"I do not support the bill. We are looking, though, for the best path forward," Mr Cantor declared following a meeting with his party's rank and file.
The legislation is aimed at averting the looming cliff by stopping most tax hikes and across-the-board spending cuts that had been due to begin with the New Year.
Rejection by the Republican-controlled House would mean that any fiscal deal would have to start all over when a new Congress, with dozens of new members, is seated on Thursday.
The largely tax-focused legislation under discussion on Tuesday was only part of the grand deal President Barack Obama had hoped to make to address the country's chronic deficit spending.
Economists, who had warned that the combination of higher taxes and deep spending cuts taking effect at the start of the year would spin the country back into recession, were warning that even a limited agreement could still dent economic growth.
The deal under consideration Tuesday tackled one of the most sensitive issues - higher taxes.
The measure would be the first significant bipartisan tax increase since 1990. It would prevent taxes from going up on the poor and middle class but would raise rates on households making more than \$450,000 (£277,000) a year.
It also would also put off for two months more than \$100bn (£62bn) in automatic spending cuts that were set to hit the Pentagon and domestic programmes starting this week.
It would also extend unemployment benefits for the long-term unemployed.
The measure cleared the Democrat-controlled Senate hours after Vice President Joe Biden and Senate Republican Leader Mitch McConnell, veteran negotiators, sealed a deal.
Mr Obama has praised the agreement, saying: "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."
Meanwhile, both sides are gearing up for the next legislative showdown over the need to lift the government's statutory borrowing limit of \$16.4trn (£10.1trn), which was reached on Monday.
The Treasury will now take extraordinary measures to keep the government afloat for an undisclosed period of time until the ceiling is raised. Republicans are already demanding spending cuts in return.