A vote is scheduled in the U.S. Senate on Sunday that White House and Congressional officials hope could pave the way for an end to the weeks-long deadlock on an agreement to raise the government’s borrowing authority.

Tuesday is considered the deadline for Congress to increase the debt limit above the current $14.3 trillion and prevent the federal government from defaulting on its debts.

Intense negotiations took place on Saturday, a day after a Republican proposal that was approved by the House of Representatives was rejected by the Democrat-controlled Senate.

The Republican leader in the Senate, Mitch McConnell, told reporters Sunday, “I think we’ve got a chance of getting there” on a deal to increase the debt limit while cutting government spending. A White House official was quoted as saying a deal was “not there yet.”

There is concern in financial markets over a possible U.S. government default that would threaten an already fragile economy. Last week U.S. stock markets suffered their worst losses this year, and the value of the dollar slumped.

Past increases in the debt limit have been routine, but Republicans in Congress have demanded huge spending cuts with no increase in taxes as a condition for approving an increase this time.

Republicans had been calling for an immediate short-term agreement and a revisiting of the debt ceiling debate next year, while Democrats were pushing for a plan to cover the nation’s borrowing needs past the presidential and congressional elections in November 2012.