* May insists extension of "months" just an option
* EU offered extra year to ease Irish problem
* EU has long assumed transition must be longer
* May faces criticism at home over the idea (Adds further May comments, other EU leaders)
BRUSSELS, Oct 18 (Reuters) - British Prime Minister Theresa May and other EU leaders voiced renewed confidence on Thursday that they could secure a Brexit deal, saying they were working hard to overcome the very same hurdles that only days ago brought the talks to a halt.
Less than six months before Britain quits the EU in its biggest shift in policy for more than 40 years, the two sides are at odds over how to deal with their only land border, between the British province of Northern Ireland and Ireland.
The problem centers on a so-called backstop - an insurance policy to ensure there will be no return to a hard border on the island of Ireland, a former focal point for sectarian tensions, if a future trading relationship is not in place in time.
To try to unlock the talks, May had earlier signaled she would consider extending a so-called transition period "for a matter of months" after Britain leaves the EU in March, a move her critics called a betrayal but one which the bloc welcomed.
Extending the transition period could mean that if a future partnership is not ready, a backstop, which so far has been unpalatable to the British side, would not have to be triggered. But even an extension would not get rid of the EU's insistence that such a backstop must be agreed to secure a deal.
For now, both sides seemed to be happy to kick any solution to that problem a little bit further down the road.
"We are all working, we're intensifying the work on these issues that remain," May told a news conference after a two-day summit in Brussels had ended.
"What I've had from leaders around the table ... since I arrived here in Brussels yesterday is a very real sense that people want that deal to be done."
"I am confident that we can achieve that good deal."
EU leaders also voiced a new optimism after the last summit in the Austrian city of Salzburg ended in acrimony, with May irritated that the EU had criticized her Brexit plans in a particularly damaging way.
EU Council President Donald Tusk described the mood as being much better than Salzburg. "What I feel today is that we are closer to the final solutions and the deal," he told a news conference.
Jean-Claude Juncker, EU Commission president, said: "It will be done."
But behind the positive noises, one side will have to compromise to find a way to, as German Chancellor Angela Merkel said, "to square the circle" of the Northern Irish border.
On arriving at the second day of the summit, May said a "further idea" had emerged, an idea to extend the transition phase beyond December 2020 that caused uproar among some Brexit supporters back in Britain.
May and a senior British government official tried to play down the significance of the consideration of such an extension, saying it had only "come up in negotiations in recent days" and that it was one of several options to help move the talks on - something London desperately wants to see.
And a French official said any extension would come with conditions attached - it would not be automatic, it would be decided closer to the time and that it would have to be agreed by the leaders of the 27 other EU nations, probably unanimously.
Irish Prime Minister Leo Varadkar summed it up by saying big gaps remained between the two sides.
"A lot of things have been agreed but there are still big gaps both in terms of the shape of the future relationship and also the protocol on Northern Ireland and Ireland and the backstop," he told reporters.
(Additional reporting by Philip Blenkinsop, Michel Rose, Gabriela Baczynska in Brussels and William James, Kylie Maclellan and Andrew MacAskill in London, Conor Humphries in Dublin; Editing by Alastair Macdonald and Alison Williams)