* EU leaders met late on Thursday in Estonia
* Informal gathering follows fizz of initiatives
* Macron speech on Tuesday admired but many wary
* Britain's May attends after keynote Brexit speech (Adds EU source on outcome of dinner)
TALLINN, Sept 28 (Reuters) - German Chancellor Angela Merkel praised French President Emmanuel Macron's ambitions for the European Union on Thursday and said his ideas could be the foundation for "intense" Franco-German cooperation on the future of Europe.
Meeting before a summit dinner of EU leaders in Estonia, Merkel and the recently elected Macron are keen for the bloc's founders to drive it forward in the wake of Brexit, though the losses she suffered in winning a fourth term on Sunday mean Europe's leading politician faces uncertain support at home.
On Tuesday, Macron outlined bold proposals for a European renewal, calling for the European Union to work more closely on defence and migration and for a euro zone budget. He urged his peers to put European vision above national interests, saying in his address that he had "no red lines, only horizons".
However, he faced a cautious hearing when he expounded his ideas to fellow EU leaders over dinner in Tallinn. "European horizons drawn. Important to avoid mirages in the desert on the way," Lithuania's no-nonsense President Dalia Grybauskaite tweeted while the discussion was still going on.
The informal get-together was arranged on the fly before a "digital summit" on issues ranging from data and cybersecurity to raising more tax from online firms.
It had no set agenda and could range widely, even allowing for Prime Minister Theresa May to pitch her ideas on Britain's looming exit from the European Union. But diplomats said its focus will be on the fizz of new initiatives, notably from Macron.
An EU source said there was a "strong and shared willingness to maintain the unity" and that the European Union should be "open to address new ideas" while continuing to work to deliver concrete results for citizens.
A French presidential source said France was not trying to impose its ideas on its partners but to show them that they were in their common interest and recognised that some needed time to reflect.
"The idea is not about forcing people to give a binary response. France cannot force things," the source said, adding that Paris hoped leaders could agree on a way to work on the ideas in the coming weeks before an October summit in Brussels.
The Estonia meeting comes four days after a German election that has raised the prospect of months of tough coalition talks for Merkel, the most influential EU leader, and two days after Macron's rallying cry for deeper integration of national economies.
"Macron has stolen the show," one senior EU official said of the dinner debate.
Many admire the youthful new French president's energy and oratory after years in which Paris, long a driving force of the European Union, has appeared bereft of self-confidence.
Merkel told reporters before a meeting with Macron that his ideas could be the basis for "intense" Franco-German cooperation.
"As far as the proposals were concerned, there was a high level of agreement between German and France. We must still discuss the details, but I am of the firm conviction that Europe can't just stay still but must continue to develop," she said.
However, she may find it hard to commit Berlin to much as she has barely started the process of building what is likely to be a three-way coalition government.
Macron was also likely to face polite but firm resistance at the dinner to his calls for a substantial pooling of national budgets in the euro zone and a possible breakaway by the wealthy, western states into a deeper monetary union.
The French presidential source recognised that euro zone matters would not be easy and that it might not be possible to have agreement on all items, although Merkel had not earlier on Thursday ruled anything out.
Eastern European leaders are cautious about the risk of new cleavages on the continent leaving them behind, while there are plenty, like Dutch Prime Minister Mark Rutte, a moving spirit behind the Tallinn dinner, who are sceptical about more financial burden-sharing before southern neighbours -- including France -- put their own national budgets on a sounder footing.
Brussels diplomats had been a shade nervous about the leaders being left, unscripted, to their own devices at a time when not only Macron but EU chief executive Jean-Claude Juncker and others have been delivering a stream of ideas. These focus on how the bloc, emerging from a slump and a series of crises, can reinforce itself in the wake of Britain's departure in 2019.
Summit chair Donald Tusk moderated the discussion which went on past midnight and the EU source said he would "consult with his colleagues in the coming two weeks and propose how to take the work forward."
May also attended the dinner, despite Britain's increasing isolation as it prepares to quit the bloc in 18 months.
May arrived with a better sense of whether her keynote major Brexit speech last Friday has succeeded in unblocking talks in Brussels on Britain's divorce package.
The chief EU negotiator, Michel Barnier, praised on Thursday a "new dynamic" to Brexit negotiations created by concessions made by May although progress was still not sufficient to allow discussions on a transition period after Brexit or on future trade relations.
EU officials said she should not expect direct feedback in Tallinn from the other leaders. But she was expected to talk to some of them individually as she pursues her quest for agreement to open talks on close ties with the bloc after Britain leaves.
The EU insists that cannot happen until "significant progress" is made on divorce terms -- notably how much Britain owes. Her speech in Florence has, so far, averted a stalemate, EU negotiators say, opening the way for some positive movement. (Writing by Alastair Macdonald, additional reporting by Philip Blenkinsop, Julia Fioretti, David Mardiste and Thomas Escritt; Editing by Matthew Mpoke Bigg, Hugh Lawson and Lisa Shumaker)