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UPDATE 7-Britain's May wins Brexit reprieve, faces tough weeks ahead

* EU says Britain must address divorce bill to unlock talks

* May walks political tightrope in run-up to December summit

* Some Brexiteers would prefer she walk away from talks (Adds Merkel, Macron, Tusk, Juncker comments)

BRUSSELS, Oct 20 (Reuters) - British Prime Minister Theresa May won a modest reprieve in stalled Brexit talks on Friday, with European Union leaders signalling their readiness to move the negotiations forward in the coming months.

But despite a more positive tone, a weakened May now faces a delicate political balancing act as she tries to meet EU demands for more concrete pledges on Britain's divorce bill without stoking a backlash from Brexit campaigners at home, some of whom would prefer she walk away from the talks.

EU leaders said at a summit in Brussels that they would begin preparations to move into "phase two" of the Brexit negotiations in December, a step forward that would allow London to discuss its future trade relationship with the bloc.

Yet they also made clear that May would have to move between now and the end of the year on settling a financial bill that EU officials have estimated at around 60 billion euros.

"I think it is very clear what additional steps need to be taken," German Chancellor Angela Merkel told a news conference at the end of the summit, saying movement on the financial settlement was crucial for progress in December.

French President Emmanuel Macron was tougher, saying the two sides had not yet completed even half of the work on the financial settlement and accusing Britain of "bluffing" by using the media to suggest there could be no deal.

"A lot is in the hands of Theresa May," he said. An EU official said it took just 90 seconds for the 27 other leaders to adopt their Brexit conclusions at the end of the meeting, underlining how united they are.

May has said she cannot provide a specific financial pledge until she knows the shape of the future relationship. The EU is insisting that the two sides agree on an exit bill, the rights of EU citizens in a post-Brexit world and Irish border issues before delving into future ties.

Asked whether she had improved an offer of about 20 billion euros, May said she had repeated commitments she made in a speech in Italy last month, when she said the bloc would not be out of pocket when it came to its budget which runs until 2020.

"What I made clear to my EU counterparts in relation to financial contributions... is that nobody need be concerned for the current budget plans ... and that we will honour the commitments that we have made during our membership," she told a news conference before returning to Britain.

"Now there has to be detailed work on those commitments... we are going through them line by line and we will continue to go through them line by line."

NO MOVEMENT YET

The final text from the EU-27 read: "The European Union ... notes that, while the UK has stated that it will honour its financial obligations taken during its membership, this has not yet been translated into a firm and concrete commitment from the UK to settle all of these obligations."

Still, the leaders held open the hope of reaching a deal at the next regular summit in December. And in a move that could save weeks of delay, they ordered EU negotiators to start preparing for what Brussels will want in a transition period.

Uncertainty over the final shape of a Brexit deal has unsettled businesses on both sides of the Channel.

Terry Scuoler, CEO of the British manufacturers' association (EEF), welcomed the warmer words from EU leaders, but said industry needed "more than a hint of progress".

All the EU leaders worked hard to strike a positive tone at the summit after May used a dinner late on Thursday to appeal to the other 27 leaders to help her silence critics at home by offering a signal that the talks would move on.

Many were upbeat and May struck up an animated and friendly conversation at the beginning of the summit with Merkel and French President Emmanuel Macron, who are both seen in Britain as hardliners in the talks.

That might be enough for May to calm immediate concerns at home after Brexit campaigners urged the British prime minister to signal her readiness to leave the EU without a deal and to rely on World Trade Organization rules.

But she faces a major dilemma as she gears up for the next EU summit in December.

"The next eight weeks will be the most challenging for ... Theresa May and the most consequential for Brexit," said Mujtaba Rahman, an analyst at Eurasia Group.

The chair of the summit Donald Tusk said there would need to be a more positive narrative to reach a Brexit deal in December - a sentiment echoed by European Commission President Jean-Claude Juncker.

"We have some details but we don't have all the details we need," he said. "It's not my working assumption that we'll have no deal." (Additional reporting by Gabriela Baczynska, Jan Strupczewski and Alastair Macdonald; Writing by Liz Piper and Noah Barkin; Editing by Toby Chopra)