Britain is now due to leave the European Union in a week, but May has been forced to seek more time after Britain's parliament failed to approve a withdrawal agreement.
The prime minister has proposed that if UK MPs approve a deal in time, the UK should be able to leave before European Parliamentary elections on 23 May.
"Because Parliament has made clear it will stop the United Kingdom leaving without a deal, we now have a stark choice: leave the European Union with a deal or do not leave at all", she said.
A long extension would "risk giving the keys of the EU's future to a Boris Johnson or a Michael Gove", warned Guy Verhofstadt, the European Parliament's interlocutor on Brexit, referring to the two MPs who campaigned vigorously to leave the EU, and who could push Brexit in a very different direction to Ms. May.
The Friday talks ended just hours after May wrote to Donald Tusk, the European Council president, to ask to delay Brexit until June 30, this year so that British MPs can agree a withdrawal deal.
Some, like French President Emmanuel Macron, have said they want to hear a clear reason from May as to why Brexit should be delayed yet again - a move that would add to uncertainties weighing on business across the bloc.
Mrs. May has already requested an extension to the end of June but this was rejected at a summit last month. They voted it down for a third time last week.
The Europeans would prefer that Britain not take part in the European Parliament elections if it is going to leave.
Hammond said the government had no red lines in the talks.
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We've won it five times , it's not too many in the history. "We started really well". But I can not compare us with what United have done.
May's team is now holding negotiations with leaders from Britain's main opposition Labour Party in a bid to secure enough votes to push through her deal on a fourth attempt.
Tusk is understood to prefer a year-long extension, which could be shortened if and when the United Kingdom ratifies a withdrawal deal, the BBC reported. They believe it would stop a new Tory leader, such as Boris Johnson, shifting to a harder Brexit once May has been replaced.
May badly needs evidence of a viable divorce strategy to persuade the other 27 European Union leaders at a summit next Wednesday to grant a delay, preferably on her preferred departure date.
All 27 nations have to agree unanimously on an extension.
But EU sources said Mr Tusk is recommending a longer postponement of one year, with a break clause in the case of earlier ratification, in a so-called "flextension" deal.
On Saturday night Downing Street said discussions with Labour to find a Brexit compromise that could pass through parliament before Wednesday's European Union summit were "ongoing" at a technical level, but declined to be drawn on whether there were any plans to hold votes tomorrow or Tuesday, before May heads to Brussels.
The talks appeared on the verge of collapse on Friday night after shadow Brexit secretary Sir Keir Starmer said the government was not "countenancing any changes to the actual wording of the Political Declaration", the blueprint for the future UK-EU relationship.
After weeks of Brexit deadlock, with Theresa May's withdrawal agreement failing to win a majority in the House of Commons twice and once more without its political declaration, with not one single motion managing to muster a majority in a series of indicative votes, the government and opposition engaged in talks to try to end this stalemate.
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