Even before Bercow's ruling, May faced a struggle to reverse the huge margins of defeat for the Brexit divorce agreement in Parliament.
Mr Barnier warned that the current withdrawal agreement was "the only treaty available" and remains the "only possible basis" for the United Kingdom to leave with a deal.
The 2016 referendum, which produced a 52-48 percent vote to leave, exposed deep divisions and has fuelled soul-searching about everything from secession and immigration to capitalism and British identity.
MPs will be given an opportunity in the House of Commons on Monday to debate how the process should go forward.
"If not, what would be the objective and the outcome of an extension?" he asked.
"In any case, the European Council will have to assess what is in the best interests of the EU".
Meanwhile, the European Council has adopted a series of contingency measures in the event of the United Kingdom leaving the EU without a deal.
But after Bercow's bombshell declaration that Parliament can only vote again if the deal is "fundamentally different", May faces the prospect of requesting a much longer delay.
One cabinet source said, "the current position is untenable - if they just accept a long extension the party will split".
Sterling dropped below $1.32 with investors saying it had damaged May's already sagging prospects of getting her European Union withdrawal agreement ratified by parliament before March 29.
I never thought anyone could betray us more than May with her BRINO, Brexit in name only deal but Bercow has managed that.
It comes after Downing Street admitted that the United Kingdom is facing a political "crisis" as Prime Minister Theresa May prepares to request an extension to the withdrawal process.
The Prime Minister is set to take her deal back to the Commons on Tuesday or Wednesday, after it was defeated earlier this week by 149 votes, and there is renewed hope in government that it could squeak through.
At home, a new vote on UK Brexit deal has been blocked by the Speaker of the House of Commons, John Bercow.
He told reporters in Brussels: "If there is going to be a request for a long extension of Article 50 by the United Kingdom then there will need to be a very persuasive plan to go with that to explain why that's needed and how they will use the time to conclude the outstanding issues that haven't been able to be agreed in London in the context of the Brexit process".
British Foreign Secretary Jeremy Hunt said he saw "cautious signs of encouragement" that the deal might make it through Parliament this week.
Anna Soubry, a lawmaker who left the Conservative Party over its handling of Brexit to join the new Independent Group, told Parliament: "This has to be unprecedented, the crisis that's now upon the country".
The UK can ask for a delay to the Brexit date, and May needs to ask fellow European Union leaders formally to grant that extension.
Bercow said he allowed the Commons to vote for a second time on May's deal because it was a substantially different proposal, with new legal texts for MPs to consider.
"Events yesterday tell you that situation had come to pass", he added. "We have taken big steps towards the United Kingdom", he said. Several Brexit-backing Tory MPs have long accused him of being anti-Brexit and many MPs have chaffed under his colourful rebukes and withering shouts of "order" and "calm yourself".
Roth said it was time "to put an end to the games".
He added: "As long as people think we can get to no deal, they will vote the deal down".
French President Emmanuel Macron has hinted he could block the Brexit delay if the United Kingdom does not have a valid reason to justify it.
He is also expected to meet with Mrs May's de-fact deputy David Liddington, who is in the Belgian capital for talks with Mr Barnier.
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