Facing crunch time, Prime Minister Theresa May on Tuesday offered to move her stated position on Brexit widely seen as intransigent, to seek another extension to Article 50 and hold talks with Labour leader Jeremy Corbyn to resolve the logjam.
Corbyn welcomed May's move and said he would be "very happy" to meet her.
May announced her gambit after seven-hour long, but inconclusive, talks with her cabinet ministers in London on Tuesday and after MPs rejected a series of Brexit proposals on Monday.
In Brussels on Tuesday, the EU's chief Brexit negotiator Michel Barnier warned again that if the United Kingdom still wants to leave the bloc in an orderly manner, the current Brexit deal is the only way.
The offer of talks allows Labour to push its own Brexit preferences, which include a permanent customs union with the EU, and possibly a referendum on a deal.
Pressed again on a People's Vote, Barclay said: "I don't think we should have a second referendum because it takes us back to square one, is my personal view".
If she can't get her withdrawal agreement through Parliament before a summit of European Union leaders on April 10, she will face a perilous set of options.
If May gets the deal through parliament, Brexit can be postponed until May 22.
May will also be meeting with the first ministers of Scotland and Wales on Wednesday in her attempt to secure broad national support for her deal.
According to May's plan, if she and Corbyn can somehow hammer out a compromise, the government would then present the package to House of Commons for a vote. Over half of her lawmakers voted last week to go for a no-deal Brexit, to the dismay of British business.
The EU has repeatedly refused to renegotiate the Brexit divorce deal it agreed with May last November after nearly two years of negotiations but has said it could revise the accompanying political declaration on future relations.
The United Kingdom was supposed to leave the European Union last Friday, but three years after Britons backed leaving the bloc in a referendum, it is still unclear how, when or even if it will do so. "Being prepared for no deal does not mean that there will be no disruption", he said.
He accused the party of being unwilling to agree to compromise over Brexit.
If the deal was rejected, London must choose between a no-deal Brexit and a long delay, he said.
"We still have a little bit of patience", Barnier said, but urged British legislators and the government to take a more realistic approach as soon as possible.
"If the government won't act urgently then Parliament has a responsibility to try to ensure that happens, even though we are right up against the deadline".
British newspapers reported numerous Brexiteers in her cabinet opposed the Labour outreach and instead favour pursuing a no-deal departure - though several publicly backed May's new stance.
Verhofstadt has long said that effective cross-party cooperation in the House of Commons is the best - and perhaps the only - way for the U.K.to emerge from the crisis.