Video report by ITV News Europe Editor James Mates
The president of the European Commission has said the Commons votes on Tuesday had increased the risk of a no-deal Brexit as he stressed the EU would not be renegotiating.
Following Tuesday's votes on seven amendments Theresa May had hoped to open talks again with Brussels after securing a mandate from parliament to renegotiate her Brexit plan following the massive defeat of her original deal.
But in an address in Brussels on Wednesday afternoon, Jean-Claude Juncker said "the Withdrawal Agreement remains the best and only deal possible" insisting "it will not be renegotiated".
With Brussels standing firm in their position, the prospect of securing a deal by March 29 - the scheduled date the UK is to leave the EU - looked increasingly remote.
Mr Juncker said: "Yesterday's vote has further increased the risk of a disorderly exit of the UK. We have tried everything in our power to prepare for all scenarios, including the worst."
MPs voted by a margin of 317 to 301 to back a plan - the "Brady amendment" - which requires the prime minister to replace the Agreement's controversial backstop with "alternative arrangements" to keep the Irish border open after Brexit.
Video report by ITV News Political Correspondent Romilly Weeks
Members of the European Research Group of Tory Eurosceptics swung behind the prime minister to hand her victory on Tuesday just weeks after consigning her Brexit Plan A to a record Commons defeat.
But the EU has already dismissed Mrs May's Plan B before she had even had a chance to head back to Brussels.
Mr Juncker made it clear, Ireland, as a member state, was the bloc's priority.
"This is not a game and neither is it a simple bilateral issue. It goes to the heart of what being a member of the EU means.
"Ireland's border is Europe's border and it is our Union's priority," he said.
While Mr Juncker spoke at the European Parliament, the prime minister was having a "useful exchange of views" with Labour leader Jeremy Corbyn as they met to discuss a "sensible" approach to Brexit.
She also spoke on the phone to Taoiseach Leo Varadkar who reiterated the EU and Irish line that the backstop was non-negotiable.
Mr Corbyn, who had initially refused to engage in cross-party talks with the prime minister until no-deal was taken off the table, described his meeting with Mrs May as "exploratory on the issues".
Video report by ITV News Correspondent Emma Murphy
He said he had "set out the Labour case for a comprehensive customs union with the European Union in order to protect jobs in this country".
Despite the amicable tone of the meeting, Mr Corbyn remained "suspicious" that Mrs May was trying to "run down the clock" on Brexit.
He warned: "The whole process looks like it's running down the clock by saying, well, it's either the problems and the difficulties of no deal or support a deal that's already been rejected by the House of Commons."
The prime minister's deal was rescued with help from fourteen Labour MPs who voted with the Conservatives in favour of the Brady amendment or abstained on Tuesday night.
Mr Corbyn said he was "disappointed" and said he "we will be having discussions with them."
Following the vote on Tuesday, European Council president Donald Tusk insisted the Withdrawal Agreement struck last November was the EU's final offer, a view reiterated to ITV News by Michel Barnier, who said the 27 member state's position was "clear".
The details of the revised plan were still being thrashed out, with Downing Street suggesting that the UK's position could involve a time-limit or exit clause to the backstop or swapping it for a free trade agreement, as proposed in the so-called Malthouse Compromise drawn up by MPs from the Tories' Remain and Leave wings.
Tory Brexiters appeared unconcerned at the prospect of no-deal, putting the ball firmly in Brussels' court.
Former Brexit secretary Dominic Raab told ITV News: "If the EU is willing to show the pragmatism, flexibility we've shown there's a deal to be done but if they keep taking an intransigent approach, a stubborn approach, a computer-says-no approach then we'll leave on 29th March on WTO."
Speaking to ITV News on Wednesday morning, Michael Gove added: "The prime minister got a handsome majority last night and that majority was clear.
"We do want a withdrawal agreement but the backstop will have to change and the prime minister will be negotiating with her European partners to get the very best deal for Britain."
But Labour said Mrs May had put party before country.
Shadow international trade secretary Barry Gardiner said Mrs May should have stood firm against the Brexiters within the Tory party rather than seeking a compromise from the EU.
Speaking to ITV News he said: "I hope it isn't a wasted two weeks, it will be wasted unless she changes her red lines - by sticking by them she has said to the EU 'you need to give me a solution to my problem to help keep my party together'. What she should have done is change her red lines and said to the right wing of her party 'you're going to have to compromise'."
Labour's Ian Murray, a supporter of the Best for Britain campaign for a second EU referendum, said: "The PM is going to Brussels to demand alternative unicorns to the Northern Irish backstop but can't tell anyone what these alternatives are.
"That's because there isn't an alternative. Her only aim is to look after her party at the expense of the country."
Irish Taoiseach Leo Varadkar said the Brady amendment failed to specify the details of the alternative arrangements and that such an alternative "didn't exist".
His foreign minister, Simon Coveney, said the UK's approach to Brexit was "posing real difficulties" and accused the government of ignoring assurances that the backstop would only ever be used temporarily.
He said: "There were very strong reassurances given to the British government and the British parliament both by President Junker and President Tusk in relation to the backstop, in relation to trying to prevent it ever being used, in relation to giving assurances if it ever was used it would be temporary.
"But unfortunately they were not accepted. So the British parliament has now asked the PM to look to renegotiate the withdrawal agreement which is not open for renegotiation."
Following Tuesday night's votes in the House of Commons, Mr Tusk's spokesman said: "The Withdrawal Agreement is and remains the best and only way to ensure an orderly withdrawal of the UK from the European Union.
"The backstop is part of the Withdrawal Agreement, and the Withdrawal Agreement is not open for re-negotiation."
On Wednesday the European Parliament's chief Brexit official Guy Verhofstadt expressed bafflement at Westminster's process.
Referring to Tuesday's Westminster session, which saw seven Brexit amendments of which two were passed, he said: "What needs to stop is this: an amendment with 10 votes for, then an amendment with 10 votes against, an amendment that barely pulls through, one that fails."
"That is no way to build a future relationship with the EU," Mr Verhofstadt said.
MPs supported a cross-party amendment opposing a no-deal Brexit by 318 votes to 310. But while this will place political pressure on the prime minister, it is not legally binding on the government, unlike the legislation which remains on the statute book naming March 29 as exit day.
Those determined to prevent no-deal or delay Brexit will have a further opportunity no later than February 14, when Mrs May has promised the Commons a further opportunity to vote on her plans.