Democratic and Republican lawmakers are also at loggerheads over a partial government shutdown, which is now in its third week. But congressional leaders say there has been progress in their negotiations to find a way out of the impasse. The IMF says a US debt default would trigger global economic turbulence.
On Thursday, the US must raise its $16.7tn borrowing limit so that the nation can pay its bills. In a statement, the White House said talks scheduled for Monday afternoon had been postponed to "allow leaders in the Senate time to continue making important progress towards a solution that raises the debt limit and reopens the government". The meeting was not immediately rescheduled.
According to US media, congressional leaders are discussing a deal to fund the government until January 15 and raise the debt ceiling until around mid-February.
President Obama sounded a warning as he helped out at a soup kitchen in Washington DC earlier on Monday.
"This week, if we don't start making some real progress, both the House and the Senate - and if Republicans aren't willing to set aside their partisan concerns in order to do what's right for the country - we stand a good chance of defaulting," he said. "And defaulting would have a potentially devastating effect on our economy."
Expected to meet President Obama and Vice-President Joe Biden at the White House were Senate Democratic Majority Leader Harry Reid, Senate Republican Minority Leader Mitch McConnell, Republican House Speaker John Boehner and House Democratic Minority Leader Nancy Pelosi.
On Monday evening, Sen Reid said on the floor of the upper chamber that he and Sen McConnell had made "tremendous progress" towards a deal to raise the debt limit and reopen the government.
He added: "We hope with good fortune, and the support of all of you, recognizing how hard this is for everybody, that perhaps tomorrow will be a bright day. We're not there yet."
Earlier, Sen McConnell also expressed optimism, following what he described as "a couple of very useful discussions" with the Democratic leader. A separate bipartisan group led by Republican Senator Susan Collins also met for several hours earlier in the day to discuss possible solutions, the Associated Press news agency reported.
Congressional Democrats were said to be using the looming debt ceiling deadline as leverage to target previously enacted cuts to the US government budget. Those deep military and domestic spending cuts, known as the "sequester", went into effect in January 2013 after Democrats and Republicans failed to reach a budget compromise.
Analysts say the Senate talks represent the last best hope for a debt deal before Thursday, after talks between the White House and the Republican-led House of Representatives collapsed last week.
Government and private sector analysts have warned for weeks of the dire consequences should Congress fail to reach an agreement on raising the nation's debt ceiling.
The US treasury department has been using what are called "extraordinary measures" to pay the nation's bills since the nation reached its current debt limit in May. Those extraordinary measures will be exhausted by 17 October, US Treasury Secretary Jack Lew has said.
A swathe of government services remain closed for business after Congress missed an October 1st deadline to pass a budget, with Congress unable to agree to a law to keep the government funded.
Hundreds of thousands of federal employees were sent home and government offices closed.
Republicans would not approve a new budget unless President Obama agreed to delay or eliminate the funding of the Affordable Care Act, his signature healthcare reform law of 2010. Mr Obama has refused to budge on the matter, accusing Republicans of "extortion" in using the shutdown and the nation's debt limit as leverage in negotiations.