Macron's EU vision will bolster Franco-German axis - Merkel

* 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)

By Philip Blenkinsop and Julia Fioretti

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.