In a press conference, European Council President Donald Tusk said that, until 12 April - the deadline by which the United Kingdom would have to indicate whether it would stand candidates in the 2019 European Parliament elections - "all options remain on the table".
Meanwhile, European Commission President Jean-Claude Juncker said the legally-binding reassurances over the controversial backstop part of the withdrawal deal which were agreed in Strasbourg last week have now been endorsed.
Following more than an hour of explanations that with days left until Britain might crash out she could win over lawmakers who have twice rejected her European Union withdrawal deal, Mrs May left the summit room on Thursday and the other 27 leaders conferred.
The EU has repeatedly said that it would only allow British seafood exporters tariff- and quota-free access to EU markets in return for an agreement that EU fishing fleets can continue to operate in British waters.
"If it is not decided do so by then the option of a long extension will immediately become impossible".
Sir Graham Brady, chairman of the 1922 Committee of Tory MPs, is said to have visited the Prime Minister on Monday to tell her that a growing number believe she has to quit. I sometimes have the feeling that we are in the waiting room, a bit like Waiting for Godot.
He said: 'I have just come from Westminster to get away from Brexit, so I don't think it's appropriate to answer that question'.
In a news conference later, Mrs May told reporters British MPs will have the chance next week to look at choices they face.
Ms. May said she welcomed the new deadline but she noted that Britain could still crash out of the European Union on April 12.
The televised message was described as a "low blow" by former minister Sam Gyimah.
Why Are There Two Article 50 Extension Dates?
An address to the nation in which she blamed parliament for a failure to secure Brexit appeared to irritate the very lawmakers she needs to win over next week.
"We have worked tirelessly to negotiate the Withdrawal Agreement. They have hard jobs to do", she said.
Even if the government whips up enough votes, the deal may not even be put before Parliament. Ireland's Indian-origin Prime Minister Leo Varadkar echoed the sentiment, adding that there was "openness to an extension" as nobody wants a no-deal Brexit.
On Wednesday, May wrote a letter to the European Council President, Donald Tusk, asking for Brexit to be delayed.
Mr Tusk said: "According to our poll, hell is still empty and it means there are a lot of spaces".
Businesses and economists say a no-deal Brexit would cause huge disruptions and billions in costs to the economies of both Britain and the EU.
"So I think now it's nearly like a dam bursting, because we've been held back in a sense, it's nearly like last chance saloon now".
"We will do everything in the remaining, admittedly few, days to achieve an orderly, joint solution".
Labour Party leader Jeremy Corbyn called May's continued attempts to salvage the deal and move towards a withdrawal from the E.U.in June "unacceptable and reckless".
After a meeting that stretched through the afternoon and over dinner, the bloc said Britain could postpone its March 29 departure to May 22 - if the UK Parliament approves Prime Minister Theresa May's divorce deal with the bloc next week.