Conservative Republican Sen. Tom Coburn of Oklahoma predicted rejection of the GOP proposal would doom any chance of an agreement.
"I'm certain that if this is not good enough for the White House, we will go over the fiscal cliff," Coburn told CNN Monday night, adding the GOP offer was a compromise on taxes and spending compared to previous negotiations of the past two years.
The Republican offer was signed by Boehner, House Republican leader Eric Cantor and the chairmen of committees involved in budgeting and tax issues.
All have taken part in the various rounds of negotiations with the White House and congressional Democrats since 2010, including failed deficit reduction talks, debt ceiling brinksmanship that sparked last year's unprecedented downgrade of the U.S. credit rating, and a special committee that was unable to reach agreement on a deficit reduction plan.
Obama's re-election and Democratic retention of their Senate majority last month ensured their side's participants in the fiscal cliff talks would remain the same.
The current negotiations take place amid the lame-duck session of Congress before the newly elected legislators arrive in January. Possible outcomes include an agreement now to avoid the fiscal cliff and devise a framework for further negotiations in the new Congress on a broader deficit reduction deal.
Coburn, however, worried that politics will continue to trump reason in Washington.
"I'm okay to compromise even on some of my issues if in fact we'll solve the problem," he said Monday night. "But what we have is a game being played ... for the extreme right wing and the extreme left wing in this country rather than coming together and leading and solving the problem."
The GOP proposal includes $800 billion from tax reform, $600 billion from Medicare reforms and other health savings and $600 billion in other spending cuts, House Republican leadership aides said. It also pledges $200 billion in savings by revising the consumer price index, a measure of inflation.
White House spokesman Dan Pfeiffer criticized the Republican offer for not meeting "the test of balance."
"Until the Republicans in Congress are willing to get serious about asking the wealthiest to pay slightly higher tax rates, we won't be able to achieve a significant, balanced approach to reduce our deficit," Pfeiffer said.
The White House has made clear Obama will veto any measure that fails to increase tax rates on the wealthy. However, aides have signaled a possible willingness to negotiate the specific rate increase.
Asked repeatedly Tuesday about whether Obama insisted on a return to the 1990s rates on income over $250,000 for families or $200,000 for individuals, Carney responded by saying the president would refuse to sign any measure extending the current lower rates.
In the interview with Bloomberg TV, Obama said lower tax rates for the wealthy could be negotiated as part of broader tax reform in 2013, but only after those rates increase now.
Republicans offered the plan amid pressure for a House vote -- which Boehner has so far prevented -- on the Senate measure. Democrats launched a procedural effort Tuesday to try to force a vote.
In line with his stances during his first term and reelection campaign, Obama's deficit-reduction plan would increase taxes by almost $1 trillion over 10 years, a significant portion of a $4 trillion overall deficit-reduction goal.
It also would close loopholes, limit deductions, raise the estate tax rate to 2009 levels and increases tax rates on capital gains and dividends. The Obama plan includes $50 billion in stimulus spending for programs intended to create jobs, such as repairing roads and bridges.
Republicans object to any increase in tax rates, even for the wealthiest Americans. Instead, they want additional revenue to come from broader tax reform that lowers rates but eliminates loopholes and deductions.
Democrats respond that closing some tax loopholes for the wealthy would fail to generate enough revenue to contribute significantly to deficit reduction. Coburn, however, rejected that argument.
"That's baloney," he said. "It's easy to get $800 billion out of the wealthy in this country by limiting deductions and taking away options that specifically benefit only the well-off in this country."
Experts have said failing to reach a fiscal cliff deal and devise a framework for a broader deficit reduction package to be negotiated when the new Congress is seated in January will cause economic turmoil. The non-partisan Tax Policy Center estimates that middle-class families would pay about $2,000 a year more in taxes without action.