Video report by ITV News Europe Editor James Mates
Theresa May has finished her whistle-stop tour meeting EU leaders Angela Merkel and Emmanuel Macron for last-minute Brexit talks, hoping to win their support to back her delay.
MPs also backed a Government motion for Mrs May to seek an extension of the Brexit process to June 30 by 420 votes to 110, majority 310.
But European Council President Donald Tusk has said there is "little reason to believe" the ratification process of Mrs May's Brexit deal can be completed by the end of June.
In an invitation letter to members of the EU council ahead of their meeting on Wednesday, he said: "In reality, granting such an extension would increase the risk of a rolling series of short extensions and emergency summits, creating new cliff-edge dates.
Mr Tusk continued: "The continued uncertainty would also be bad for our businesses and citizens. Finally, if we failed to agree on any next extension, there would be a risk of an accidental no-deal Brexit.
"This is why I believe we should also discuss an alternative, longer extension.
"One possibility would be a flexible extension, which would last only as long as necessary and no longer than one year, as beyond that date we will need to decide unanimously on some key European projects."
However Downing Street confirmed Mrs May and German chancellor Angela Merkel are united in agreeing "the importance of ensuring Britain's orderly withdrawal" from the EU.
"The Prime Minister outlined the steps the Government is taking to bring the Brexit process to a successful conclusion and updated Chancellor Merkel on the ongoing discussions with the Opposition," said a Downing Street spokesman.
"The leaders agreed on the importance of ensuring Britain's orderly withdrawal from the European Union."
Theresa May also held talks with the French President Emmanuel Macron in Paris, with the UK expected to leave the EU in three days' time.
The Government and Labour's negotiating teams also held Brexit talks to try and reach a compromise.
"We have had further productive and wide-ranging talks this afternoon, and the parties have agreed to meet again on Thursday once European Council has concluded," a Number 10 spokesman said.
But a Labour spokesperson said there had not yet been a "clear shift" in the Government's position in cross-party talks.
The Democratic Unionists have branded Mrs May's bid for a Brexit extension as humiliating and embarrassing.
Party leader Arlene Foster and Westminster leader Nigel Dodds both accused Mrs May of "begging" European leaders for help to break the impasse.
"The talks between the Prime Minister and the leaders of France and Germany is humiliating and embarrassing for the United Kingdom," Mr Dodds said on Tuesday evening.
"Nearly three years after the referendum the UK is today effectively holding out a begging bowl to European leaders."
Ms Foster was scathing in her assessment of Mrs May's efforts.
"It really is quite painful to watch the prime minister at the moment," she said. The DUP has propped up Mrs May's minority Government since the Tories lost their majority in 2017.
Earlier, EU chief Brexit negotiator Michel Barnier said Brussels could "improve and amend" the Political Declaration on future relations with the UK "within a few hours or days" if the cross-party talks in London produce a new plan.
While he insisted the Withdrawal Agreement was not open to renegotiation, Europe was prepared to "provide an increased level of ambition" - such as a free trade agreement - "if that is the wish of the UK".
Speaking in Luxembourg, Mr Barnier added: "We are willing to improve and amend the Political Declaration rapidly, within a few hours or days."
Referring to the on-going talks between the Government and Labour's Brexit team, Mr Barnier said Europe was ready to respond favourably to any proposal regarding a customs union, should that be an agreed way forward from the discussions.
He said he hoped the cross-party talks would allow "a new process to emerge in the House of Commons and ultimately for a new majority to emerge with regard to the Withdrawal Agreement and Political Declaration".
He stressed it was for Mrs May and the UK to spell out what they see as the purpose of any extension to the Brexit process.
Mr Barnier added: "We are attempting to give the UK this last opportunity to achieve this orderly withdrawal, it's as simple as that.
"More time might be needed or less time, depending on the pressure you wish to exert."
Meanwhile, Government aide Huw Merriman has defied the whips by addressing a People's Vote campaign event.
The Tory MP for Bexhill and Battle told an audience in London it was "seriously wrong" that he had been threatened with the sack if he explained to them why he now supported a second referendum.
Mr Merriman said he wanted to deliver Brexit: "I want the Prime Minister's deal delivered and if she feels it's a good deal then she should also want to find the best, optimal solution which now is not Parliament, it's actually putting it over to the people and saying 'we've failed as Parliament, if you want this delivered, this is it, it's down to you'."
He added that a People's Vote would end the uncertainty and "get this country through the mess we are currently in".
Ahead of Mrs May's meetings, Commons leader Andrea Leadsom said it would be "fantastic" if Ms Merkel agreed to reopen talks on the backstop.
Ms Leadsom added: "The Prime Minister is seeking a delay, that is what the House has pushed upon her and she'll be seeking a delay to June 30.
"But, as the person with the responsibility to get the legislation through, if we can get the Prime Minister's deal over the line because the EU have decided to support measures on the backstop then that would be the best possible outcome but we will have to see what happens."
In an ominous warning note, Liam Fox, the International Trade Secretary, said a customs union with the EU - one of the possibilities being discussed between the Government and Labour - would leave the UK "stuck in the worst of both worlds".
In a four-page letter to the 1922 committee of backbench Tory MPs, Mr Fox explained: "A customs union will stymie the ability to open markets around the world to the services sectors as an 80% services economy, this is a huge area of comparative advantage to the UK."
The contents of the letter suggest Mr Fox may quit the Cabinet if Theresa May pursues such a policy with Labour.
All 27 remaining EU heads of government must agree an extension if the UK is to avoid the default position of a no-deal Brexit on April 12.
After officials from the two sides met on Monday, the fresh round of talks will include Mrs May’s de facto deputy, Cabinet Office Minister David Lidington, Labour’s shadow Brexit Secretary Sir Keir Starmer and John McDonnell, the shadow chancellor.
Labour leader Jeremy Corbyn has called on the Government to be more flexible regarding red line issues in the talks.
Mr McDonnell said on Tuesday morning he would raise the issue of "going back to the people".
He said: "What we are trying to do is that we arrive at a deal which, first of all, protects jobs and the economy - we don't think Theresa May's deal does that.
"And we are also discussing the issue apparently she has raised in Cabinet as well about the issue about going back to the people as well."