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Talks to end the U.S. shutdown and extend the government's borrowing authority beyond an October 17 deadline failed to reach an agreement on Thursday, but hopes were raised that a deal might be made on Friday.
U.S. stock futures headed higher, unwinding earlier losses, after House Republicans said the president did not flatly reject their offer of a short-term debt limit plan.
(Read more: Why it's 'incredibly treacherous' to trade right now )
Earlier Thursday, House Speaker John Boehner led a delegation of Republicans to the White House for a late-afternoon meeting with President Barack Obama as the two sides groped for a way out of the latest in a string of crises.
The U.S. shutdown has lasted 10 days, undermined sentiment in financial markets and left hundreds of thousands of federal employees out of work.
Following the meeting, Rep. and Vice Chair of the House Republican Conference Lynn Jenkins, said, "Opening the government is a negotiation that will happen tonight and in the hours ahead, and we hope to have it open by Monday night."
During the meeting, Obama and congressional Republicans found no specific way forward to break their impasse over the government shutdown and extending the U.S. debt ceiling, the White House said.
The session of about an hour and a half between Obama and Republican leaders of the House of Representatives was described as a "good meeting" where Obama heard Boehner explain Republican proposals for a short-term extension of the debt ceiling.
"After a discussion about potential paths forward, no specific determination was made,'' said a White House statement. "The president looks forward to making continued progress with members on both sides of the aisle.''
Earlier the New York Times reported Obama rejected a proposal from House Republican leaders to extend the U.S.'s borrowing authority for six weeks. Futures briefly sank on the report, but rebounded as details of the meeting emerged. (Click here to track U.S. stock futures.)
(Read more: American mood worst since 2008 crisis: Poll )
Following the meeting, Rep. Paul Ryan told NBC News, "The president didn't say yes, he didn't say no. It was a useful conversation. We're now going to negotiate."
A House GOP aide said, "There was a useful conversation. The President did not say yes or no to House Republicans' offer. Both sides are continuing the discussion tonight."
House Majority Leader Eric Cantor said the White House meeting was useful and 'we expect further conversations tonight.'
"The economic impact is very little, but turn off the noise already," Peter Boockvar, chief market analyst at economic advisory firm The Lindsey Group, told CNBC. "Politics is the art of negotiation, and each side has to give something."
Hopes that lawmakers are making progress towards a deal boosted Wall Street shares on Thursday, while Asian stock markets hit a three-week high on Friday.
The dollar, which has been hurt by the gridlock in Washington, also headed up. The U.S. dollar index traded at about 80.45, up about 1 percent from an eight-month low hit earlier this week.
"I think they [lawmakers] will raise the debt ceiling and end the shutdown. They have created a crisis so that we can be saved from it," Peter Schiff, CEO at Euro Pacific Capital told CNBC.
—CNBC With Wires