Washington (CNN) -- President Barack Obama is continuing to pursue the most "significant deficit reduction package possible," White House Press Secretary Jay Carney said Thursday, the latest indication officials are keeping a variety of options open while trying to hammer out an agreement to raise the nation's debt ceiling and avoid an unprecedented default.

The president and Vice President Joe Biden met for almost two hours Thursday with Democratic leaders from the House and Senate as sources indicated the negotiations were focusing on a deal to cut $3 trillion in federal deficits over the next 10 years that would be accompanied by a debt ceiling increase.

According to the congressional aides who spoke on condition of not being identified, the possible deal remains in limbo over a disagreement on whether to extend Bush-era tax cuts for families earning more than $250,000 a year. Nothing has been agreed to yet, they noted.

The possible deal would include spending cuts expected to total $1 trillion or more agreed to in earlier negotiations led by Vice President Joe Biden, the sources said. It also would reform entitlement programs by changing the eligibility age for Medicare over time, and using a more restrictive inflation index for Social Security benefits, according to the sources.

On taxes, it would permanently extend the Bush tax cuts for families earning less than $250,000 while allowing the cuts to expire at the end of 2012 for those with income above that, the aides said. At the same time, the deal would include a commitment to reform the tax code next year, which is expected to lower all tax rates and eliminate loopholes and subsidies, the sources said.

However, House Speaker John Boehner, R-Ohio, wants the deal to make all of the Bush tax cuts permanent while keeping the commitment to tax reform, the sources said. Republicans oppose any tax hikes, and their resistance has been a major obstacle to any deal in the negotiations so far.

Some sources said the deal would work in two stages, with spending cuts and a debt ceiling increase occurring right away while entitlement reforms and tax reforms would occur later.

Earlier, Carney denied a report by the New York Times that Obama and Boehner were close to reaching a deal.

"There is no deal. We are not close to a deal," Carney told reporters. "There is no progress to report."

A spokesman for Boehner's office echoed Carney, denying any reportable progress. House Majority Leader Eric Cantor, R-Virginia, told reporters he was "unaware of any deal that has been struck."

Other signs pointed to possible movement in the talks. Carney signaled to reporters earlier in the week that Obama may now be willing to sign a short-term debt limit extension if Democratic and Republican leaders are close to agreement on a broader deficit reduction deal that includes both tax hikes and spending reforms.

Obama previously indicated he would veto any short-term extension.

Boehner huddled with some Republican freshmen after meeting with Obama on Wednesday night, and is set to hold a previously unannounced meeting with the entire House GOP caucus Friday. He told reporters Thursday that while some House Republicans wouldn't compromise, he didn't believe they "would be anywhere close to the majority."

The highly contentious negotiations -- reflecting the core ideological beliefs of both parties -- have now become a race against the clock. If Congress fails to raise the $14.3 trillion limit by August 2, Americans could face rising interest rates, a declining dollar and increasingly jittery financial markets, among other problems.

The seriousness of the overall situation was reinforced last week when a major credit-rating agency, Standard and Poor's, said it was placing the U.S. sovereign rating on "CreditWatch with negative implications." Another major agency -- Moody's Investors Services -- said it would put America's bond rating on review for a possible downgrade.

"Even if Washington did raise the debt ceiling after just a few harrowing days following a default ... we envisage that the economy could fall quickly back into recession," Standard and Poor's said in a report Thursday.

As congressional leaders searched for a compromise, the Democratic-controlled Senate planned to vote Friday on the so-called "cut, cap and balance" plan passed by the Republican-controlled House this week.

The measure would allow an increase in the debt ceiling only after Congress passes a balanced-budget amendment to the Constitution and imposes both major spending cuts and caps on future spending as a percentage of the country's gross domestic product.

"Cut, cap and balance" is widely acknowledged to have virtually no chance of clearing the Senate or overcoming a promised presidential veto. Voting on it, however, allows Republicans to demonstrate their preference for steps favored by many in the tea party movement.

Lawmakers are also continuing discussions focused on the $3.7 trillion debt reduction blueprint put forward by the "Gang of Six," a group of three Democratic and three Republican senators.

Under the group's proposal, $500 billion in budget savings would be immediately imposed, with marginal income tax rates reduced and the controversial alternative minimum tax ultimately abolished.

The plan would create three tax brackets with rates from 8% to 12%, 14% to 22%, and 23% to 29% -- part of a new structure designed to generate an additional $1 trillion in revenue. It would require cost changes to Medicare's growth rate formula as well as $80 billion in Pentagon cuts.

Obama has praised the plan, calling it "broadly consistent" with his approach to debt reduction because it mixes tax changes, entitlement reforms and spending reductions.

Congressional leaders, however, have warned that there is most likely not enough time to translate the Gang of Six plan into legislation, tie it to a debt ceiling hike and pass it by August 2. In addition, the proposal has been hit with a barrage of criticism from both the right and the left.

Conservatives have complained about some of the plan's tax changes, while liberals have warned it would cut entitlement benefits too deeply.

One huge stumbling block to a bipartisan agreement has been staunch GOP opposition to any form of a tax hike. Over the past several years, resistance to higher taxes has become a bedrock principle for most Republicans, enforced by conservative crusaders such as Grover Norquist. His group -- Americans for Tax Reform -- has sponsored a high-profile pledge to oppose any tax increase.

The pledge has been signed by more than 230 House members and 40 senators, almost all of them Republicans.

A Washington Post editorial Thursday, however, reported Norquist had told the newspaper that allowing the Bush-era tax cuts to expire -- a move worth up to $4 trillion in additional revenue -- would not constitute a violation of the pledge.

Asked to comment on Norquist's interpretation, Boehner said he believes allowing the Bush tax cuts to expire would amount to a tax hike and added: "I've never voted to raise taxes and I don't intend to."

Sen. Chuck Schumer, D-New York, said Republicans should use the opening provided by Norquist to help reach a bipartisan accord.

"This is a coded message from one of the truest believers in the Republican Party that it's time for conservatives to step back from the brink," Schumer said. "Norquist has given us a potential path forward."

Norquist himself sought to clarify his remarks, declaring in an interview with MSNBC that while allowing the Bush tax cuts to expire may not technically violate the pledge, he would still oppose such a move.

If a larger deal cannot be reached, leaders from both parties have indicated they are still willing to consider a version of a fallback plan proposed Senate Minority Leader Mitch McConnell, R-Kentucky.

McConnell's plan would give Obama the power to raise the borrowing limit by a total of $2.5 trillion, but also require three congressional votes on the issue before the 2012 general election.

Specifically, Obama would be required to submit three requests for debt ceiling hikes -- a $700 billion increase and two $900 billion increases. Along with each request, the president would have to submit a list of recommended spending cuts exceeding the debt ceiling increase. The cuts would not need to be enacted for the ceiling to rise.

Congress would vote on -- and presumably pass -- "resolutions of disapproval" for each request. Obama would likely veto each resolution. Unless Congress manages to override the president's vetoes -- considered highly unlikely -- the debt ceiling would increase.

The unusual scheme would allow most Republicans and some more-conservative Democrats to vote against any debt ceiling hike while still allowing it to clear.

McConnell and Senate Majority Leader Harry Reid, D-Nevada, are also working on two critical additions to the plan, according to congressional aides in both parties. One would add up to roughly $1.5 trillion in spending cuts agreed to in the Biden talks; the other would create a commission meant to find more major spending cuts, tax increases and entitlement reforms.

Changes agreed to by the commission -- made up of an equal number of House and Senate Democrats and Republicans -- would be subject to a strict up-or-down vote by Congress. No amendments would be allowed.

CNN's Ted Barrett, Kate Bolduan, Lisa Desjardins, Deirdre Walsh and Jessica Yellin contributed to this report.