European Council president Donald Tusk has said it is his “quiet dream” for the UK to stay in the EU after the bloc’s leaders offered to delay Brexit until Halloween.

In a dramatic night in Brussels, European leaders rebuffed Theresa May’s calls for a short extension to 30 June, instead offering the prime minister a longer extension to 31 October to find a way through the Brexit deadlock.

Mr Tusk appeared to confirm Brexiteer fears that Britain's departure from the EU was being kicked into the long grass, telling Polish media that Brexit could be avoided.

"Maybe we can avoid the UK leaving the EU - this is obviously not my role, but it's my personal, quiet dream," he told the Polish Press Agency.

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In an early-morning press conference, Mr Tusk also did not rule out further extensions beyond October.

He sent a stern message to the UK: "This extension is as flexible as I expected, and a little bit shorter than I expected, but it's still enough to find the best possible solution.

"Please do not waste this time."

Ms May said that she still wanted the UK to leave the EU "as soon as possible" and insisted that European Parliament elections could still be avoided if a Brexit deal was ratified in May.

Acknowledging "huge frustration" among voters, she said: "The choices we now face are stark and the timetable is clear.

"So we must now press on at pace with our efforts to reach a consensus on a deal that is in the national interest."

Ms May faces a furious reception when she addresses MPs later on Thursday, as she told the Commons only weeks ago that "as prime minister" she was not prepared to delay Brexit any further than her June deadline.

She also set herself up to stay on as prime minister until the winter while presiding over the long delay, which is sure to anger senior Tories who are openly jostling for the leadership.

Former Brexit secretary David Davis, who resigned over Ms May's EU exit strategy, said pressure was mounting on the prime minister to quit.

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He told BBC Radio 4's Today programme: "I think what is likely to happen is the pressure for her to go will go up.

"The pressure on her to go will increase dramatically, I suspect, now. Whether it will come to anything - who knows?"

Asked if Ms May could still be in post by Conservative conference in the autumn, Mr Davis said: "I think it is going to be difficult because by that time we will have had a European election which will become a plebiscite, really, on Brexit.

"And I suspect you will see a very successful rise of a, sort of, Brexit movement, the Nigel Farage thing, and so on.