The EU also warned that Britain faced the prospect of either a chaotic exit from the bloc or delaying its planned departure date.
Labour Party leader Jeremy Corbyn plans to tell Labour members of Parliament (MPs) that he will support a second nationwide vote on Brexit if the deal proposed by Labour is rejected in Parliament on Wednesday, according to the BBC.
Thornberry further confirmed that she, Labour as a whole, and Jeremy Corbyn would be campaigning for a Remain vote.
He added: "That's why, in line with our conference policy, we are committed to also putting forward or supporting an amendment in favour of a public vote to prevent a damaging Tory Brexit being forced on the country".
"Let me be clear, I don't want to see Article 50 extended", May said.
Labour foreign affairs spokeswoman Emily Thornberry told ITV a second referendum should present a choice between May's deal and remaining in the European Union, but the Labour spokesman directly contradicted her, saying that would be an unacceptable choice.
The move is likely to cheer many Labour Party members, who have backed calls for a so-called "people's vote".
Britain's Parliament has rejected the deal once, and May has over a month to get it approved by lawmakers before the U.K.'s scheduled departure day.
European Council President Donald Tusk said he had discussed the "legal and procedural context of a potential extension" when he met May on Sunday on the sidelines of an EU-Arab summit in Egypt's Sharm el-Sheikh.
The mechanism, known as the backstop, is a safeguard that would keep the U.K.in a customs union with the EU to remove the need for checks along the Irish border until a permanent new trading relationship is in place.
The official document released by the British government was published only after constant demands from lawmakers were made to prepare for a "no deal" Brexit scenario as the March deadline draws near.
But EU leaders insist the legally binding withdrawal agreement, which took a year and a half to negotiate, can't be reopened.
The Bank of England also forecast in November that a no-deal Brexit in March would cause the United Kingdom economy to shrink by about 8% within a year, the worst drop the country has seen since the 1920s.
Alternatively, if May can not get a deal through Parliament, the second Brexit referendum would see the option "No deal" pitted against "Remain".
Labour's Hilary Benn, who chairs the Commons Brexit committee, said talk of a referendum was still some way off as MPs would have another opportunity to vote on Theresa May's deal by 12 March.
But an amendment put forward by Labour MP Yvette Cooper and Tory backbencher Oliver Letwin, seeks to bind her government into the steps, and to bringing forward the "necessary legislation to change the exit date".
The UK triggered a two-year negotiating period in March 2017 when it stated its notice to leave the European Union under Article 50 of the European Union treaty.
Tusk said, "I believe that in the situation we are in, an extension would be a rational decision, but prime minister May still believes she will be able to avoid this scenario".
But Mrs May told the Commons that she wanted any delay to be "as short as possible". When asked repeatedly if May was going to delay, Lidington did not address the questions directly. "It doesn't resolve the issue".
The votes are not on Mrs May's Brexit deal itself, but they will show what support she can or cannot get.