Arriving for the summit, she insisted that a deal is achievable and that "now is the time to make it happen".
"Both sides want to get a deal done here, and I think we need... to allow the negotiating teams to set the pace with a view to making recommendations, hopefully by mid-November, that a new summit is necessary to sign off on a final deal".
The so-called "backstop" will apply if there is no proper EU-UK trade deal post Brexit.
Staunch Brexiteer, Nadine Dorries, urged MPs to oust May as Conservative party leader and prime minister if she accepts a longer transition period.
"There's no need to dramatize matters".
British Prime Minister Theresa May is weighing up a plan to stay tied to European Union rules for longer in a radical move created to break the deadlock in Brexit talks, according to people familiar with the matter.
European Commission President Jean-Claude Juncker briefed the leaders on the EU executive's plans for a no-deal scenario.
The secret Brussels discussion on a possible extension was revealed by European Parliament chief Antonio Tajani, after he joined the pre-dinner session between May and the EU 27 leaders.
European Union diplomats said May appeared to show a greater understanding of the EU's concerns, including Ireland's requirement for a "backstop" insurance clause that would mean there would beno hard border between Northern Ireland and the Republic of Ireland. "It doesn't resolve any of the underlying issues and keeps us in a painful Brexit holding pattern".
Both sides agree there must be no hard border, but each has rejected the other side's solution.
But the Prime Minister said she does not expect any extension of the so-called "transition" to Brexit to be needed, because she still hopes to conclude a deal on the UK's future trade and security relationship with the European Union by its scheduled end-date of December 2020.
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But he said she was "not against" the idea of accepting a one year extension in the transition period after Brexit to take the heat out of the most contentious issue - the fate of the Irish border.
And in her appeal to the leaders of the 27 other EU member states, Mrs May is said to have been "not against" the European Council's proposals of extending the transition period to three years.
May's proposal to extend this period is therefore entirely sensible, even if the realisation that such an extension is needed has come very late in the process.
This week's summit has always been billed as "the moment of truth" when agreement was needed to allow time for ratification before Brexit day in March. But after urgent talks on the Irish border on Monday ended without producing the hoped-for breakthrough, the gathering looked more like a therapeutic bonding session than an occasion to celebrate.
Bertie Armstrong of the Scottish Fishermen's Federation said: "The EU must not be permitted to misuse the time when the United Kingdom is in the Brexit waiting room to try to place conditions on the return of fish stocks that are rightfully ours".
However, an extension does not in and of itself solve May's problem, which is that she has yet to fully decide, let alone agree, what sort of relationship she wants to secure during that additional transition period.
As EU leaders gathered yet again in Brussels yesterday, they conceded that hopes of a deal are again being pushed back, this time perhaps until December.
It was a gruelling House of Commons experience for May, who was grilled by all sides of the Brexit argument and reminded of how many factions she must please to have a chance of getting a Brexit deal through Parliament.
'It refers to the assurances given by the government put before parliament to approve a deal would be amendable.
"Today we do not know what they want", she said.
Upon her arrival at the European Union summit, May told reporters, "We have solved most of the issues in the withdrawal agreement".