The Democratic and Republican leaders in the Senate held direct talks for the first time in weeks, but there is little sign of any breakthrough, correspondents say.
The shutdown began when Congress missed the October 1st deadline to pass a budget.
The US faces another deadline on Thursday to raise its debt limit.
If a deal is not reached by then, the US faces potential default, a prospect which has caused alarm both domestically and abroad.
Senator Dick Durbin, a Democrat, said the aim was to reach a deal on extending the debt limit before markets reopen on Monday.
The talks between Senate Majority Leader Harry Reid and Senate Republican leader Mitch McConnell earlier on Saturday represented the first face-to-face meeting between the two since July, the New York Times reported.
“The conversations were extremely cordial but very preliminary of course – nothing conclusive, but I hope that our talking is some solace to the American people and to the world,” said Harry Reid.
“We had a good meeting,” said Mitch McConnell, without elaborating.
Harry Reid then went to the White House for talks with President Barack Obama.
But he rejected a plan put forward by Republican Senator Susan Collins to allow the government to increase its debt limit until January 31st, 2014.
Democrats have a majority in the Senate, but could not muster enough support to advance a proposal to lift the debt ceiling there.
Talks between House Republicans and the White House had collapsed earlier.
Republicans have refused to pass a new budget unless President Barack Obama agrees to delay or eliminate the funding of the healthcare reform law of 2010.
Hundreds of thousands of federal employees have been sent home as a result of the shutdown.
The White House has repeatedly said it would not undermine the law, known as Obamacare, nor negotiate over larger budget matters, until Republicans vote to end the threat of default.
It has also rejected a short-term deal over the debt limit.
“It wouldn’t be wise, as some suggest, to just kick the debt ceiling can down the road for a couple of months, and flirt with a first-ever intentional default right in the middle of the holiday shopping season,” President Barack Obama said.
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