Time is running out for an agreement on a short-term deal to avert the fiscal cliff and set up further negotiations in the new Congress that convenes in January or a more comprehensive approach that both sides say they want.

Without a deal, everyone's taxes go up and the government faces deep spending cuts, including for the military. However, the administration has signaled it can delay some of the effects to allow time to work out an agreement next year.

The House is supposed to end its current session on Friday and go home for the year, but Majority Leader Eric Cantor made clear the chamber would stay later than planned.

"We are going to stay here right up to Christmas Eve, throughout the time and period before the New Year, because we want to make sure we resolve this in an acceptable way for the American people," Cantor said Wednesday.

With a deal needing time to be written into legislation and approved by Congress, the working deadline for an agreement this year is around Christmas Day at the latest.

"You`re talking about having to have something done by Christmas, but the sooner the better," Rep. Chris Van Hollen, D-Maryland, told MSNBC on Tuesday.

Polls consistently show the public favors Obama's stance in the negotiations.

Nearly half of Americans -- 49%--say they approve of the president's handling of the talks, compared to 25% who say Boehner is doing a good job, according to a new ABC News/Washington Post poll released Wednesday.

Meanwhile, a new Bloomberg National Poll indicated that nearly two-thirds of respondents, including nearly 50% of Republicans, believe Obama's re-election gave him a mandate to seek higher taxes on the wealthy.

Also Wednesday, an administration official said the Business Roundtable --an association of CEOs of major corporations -- has dropped its opposition to raising tax rates on the top two tax brackets after lobbying by the White House.

Despite Boehner's adamant stance against such a tax rate hike, some cracks have appeared in his party's anti-tax facade, prompting a conservative backlash.

Sen. Lindsey Graham, R-South Carolina, promised the newly re-elected Obama "one hell of a fight" next year if the president forces through his plan for high-income earners to pay more taxes without agreeing to substantive steps to reduce the nation's chronic federal deficits and debt.

"There will come a time in February and March where we have to raise the debt ceiling," Graham said Tuesday on "Piers Morgan Tonight" on CNN. "I will not raise the debt ceiling ever again until we get significant entitlement reforms, because if we don't reform entitlements, we're going to become Greece."

Pelosi, however, argued Tuesday that more than $1 trillion in spending cuts agreed to by Congress in the past two years should be counted toward deficit reduction in the current negotiations.

Both sides call for eliminating tax deductions and loopholes to raise more revenue, but Obama also demands an end to the tax cuts of 2001 and 2003 for the top brackets.

Republicans oppose the return to higher rates, saying it will inhibit job growth because small business owners declare their profits as personal income and therefore would face a tax increase.

In response, Obama and Democrats note that their plan -- already approved by the Senate and needing House approval to be signed into law by the president -- affects just 2% of taxpayers and 3% of small business owners.

While Republicans argue those small business owners account for about half of all business income, Democrats say that's because they include law firms, hedge funds traders and other high-income operations.

Retiring Sen. Kent Conrad, D-North Dakota, predicted Tuesday on MSNBC that a deal would get worked out in a week's time.

"It would be wise on their part not to come too quickly with a deal because that would give all the interest groups a chance to get organized and try to kill it," Conrad said. "And we know that on the right, on the left, special interest groups are just salivating at the chance to attack any agreement because, look, any agreement is going to have controversy attached to it."