* New optimism about finding compromise on debt limit
* No deal yet, large part of government still closed
* Markets encouraged by bipartisan talks
By Richard Cowan and Thomas Ferraro
WASHINGTON, Oct 11 (Reuters) - President Barack Obama and congressional Republican leaders worked to end their fiscal impasse on Friday, but struggled to strike a deal on the details for a short-term reopening of the federal government and an increase in the U.S. debt limit.
One day after the first signs of real movement in the standoff, both sides searched for an agreement that would get federal workers back on the job and extend the government's borrowing authority past the Oct. 17 limit.
A lot of the details of the negotiations are unknown at this point. Lawmakers are expected to work throughout the weekend, with the goal of finalizing a deal by early next week.
House of Representatives Republicans on Friday waited for a White House response to their latest offer, a senior aide said, without providing details. On Thursday, Republicans floated a plan that would provide for a short-term debt limit increase lasting about six weeks.
Republicans also had dangled the possibility of a quick reopening of government if there was a commitment by Obama to broader deficit reduction steps. Obama had asked for a longer debt limit increase with no conditions attached.
"We are awaiting a response from the White House. If he (Obama) signals back that he is willing to sign a short-term deal" on the debt limit, Republicans will move to re-open the government, said Republican Representative Tom Cole of Oklahoma.
At a White House meeting with Senate Republicans on Friday, Obama expressed concerns that the debt-limit extension was too short and talked about the need for new revenues as part of any long-term deficit reduction plan, Republican Senator Orrin Hatch of Utah said.
Hatch said he left the meeting feeling the fiscal fight will still be a "difficult experience."
A handful of House Republicans expressed optimism now that talks have started with the White House.
"We've been very good about this, the White House has been great about it overnight to say let's stay out of all the details as we talk through all the different options," Oklahoma Republican Representative James Lankford said on CNN.
California Republican Representative Howard McKeon, chairman of the House Armed Services Committee, said the White House and Republicans agreed to focus on moving forward.
"There's no sense dwelling on the past. Mistakes have been made. But at least now we're talking, and it's time to move forward. It's best for the American people," McKeon said on MSNBC.
The newfound optimism bolstered markets. U.S. stocks edged higher in choppy trading on Friday, a day after their biggest rally in more than nine months.
Time was running short, with the partial government shutdown in its 11th day and less than a week to go before the Treasury Department exhausts its ability to borrow money to pay the government's bills.
'STOP THAT TRAIN WRECK'
Any deal that is struck by leaders could face a revolt from rank-and-file conservatives in both the House and Senate, risking a potential U.S. default that Obama and economists have warned could lead to economic chaos.
Texas Senator Ted Cruz, a Tea Party favorite who has been a leader of conservatives demanding delays or defunding of Obama's healthcare law before they will approve a budget deal, took a hard line at a conference of conservative activists.
In a speech frequently interrupted by hecklers but warmly embraced by the smaller-government Tea Party faithful, he said the country must "stop that train wreck, that disaster, that nightmare that is Obamacare."
The impending fiscal deadline is particularly problematic in the Senate, where procedural delays can slow legislation for up to a week - about the time remaining before borrowing authority runs out, according to Treasury Secretary Jack Lew.
Senate Democratic Leader Harry Reid will hold a vote on Friday or Saturday on a measure giving a one-year debt ceiling increase without conditions. It is expected to be blocked by Senate Republicans. The chamber may then move quickly on a shorter time frame, even if it is not Democrats' first choice.
Reid on Friday publicly criticized Republican calls for a shorter extension of the borrowing authority.
"We do not believe a six-week delay of a catastrophic default is enough to get the economy the confidence it needs," Reid said on the Senate floor.
The movement from Republicans on Thursday came after a flurry of new opinion polls showed them taking the blame for the government shutdown, that began with the start of the new fiscal year on Oct. 1.
A new NBC/Wall Street Journal poll showed Republican Party favorability ratings at an all-time low of 24 percent and Democrats with an eight-point lead on voter congressional preference heading into next year's mid-term elections.