Parliament will vote on different Brexit options on Monday and then May could try one last roll of the dice by bringing her deal back to a vote in parliament as soon as Tuesday.
May could table it together with an amendment submitted last week by the Labour MPs Gareth Snell and Lisa Nandy allowing parliament a greater say in the next stage of the Brexit negotiations.
The two proposals would see Britain remain in closely tied to European trade rules and tariff regimes.
Labour is also backing a third plan today, dubbed "common market 2.0", which would accept May's divorce terms but require her to negotiate a new European Union customs arrangement and membership of the European Union single market. A Deltapoll on Sunday found support for Opposition Leader Jeremy Corbyn's Labour Party at 41 per cent, with the Tories on 36 per cent - potentially producing a minority Labour government.
The House of Commons is debating four rival options to May's three-times rejected Brexit deal and politicians will vote on them on Monday evening.
Defense Minister Tobias Ellwood urged fellow Conservative lawmakers to compromise to ensure an orderly Brexit.
A proposal for a confirmatory referendum on any deal got the most votes, but was defeated 292-280.
Complicating the equation is the staunch opposition of May's erstwhile allies in the DUP, which refuses to back her Brexit withdrawal deal because it treats Northern Ireland differently than the rest of the United Kingdom, and the Scottish National Party, which wants to remain in the EU.
Ms Thornberry also said "it looks like the time may come" for another attempted no confidence vote in the government.
Softer Brexit: Parliament considered several measures that would enable a softer Brexit.
Attorney General Geoffrey Cox speaks in the House of Commons ahead of the Brexit debate
May's spokesman, James Slack, Slack rejected speculation that the government could take drastic action, such as asking Queen Elizabeth II to suspend Parliament or getting her to refuse to sign legislation.
Chief executive John Lundgren said: "We are operationally well prepared for Brexit".
When asked if Labour was a Remain party, Ms Thornberry replied: "In our hearts we want to remain but we have to square that with democracy".
Cabinet ministers will meet tomorrow to discuss the outcome of the ballots, and could still decide to hold a fourth vote on May's deal on Wednesday or Thursday.
The range of choices, and lack of consensus, reflect a Parliament and a government deeply divided over how - and whether - to leave the EU.
"Brexit is a shitshow, I'll say it quite undiplomatically", he said.
Maier also wrote that Britain's political paralysis is "making it hard to win support for finely balanced investment decisions that in the end have an impact on U.K.jobs, innovation and the competitiveness of our activities here".
Under a deal reached with the European Union last week, Britain is now due to exit the bloc on April 12, which leaves no time to host a general election.
Mr Cameron tried to portray himself, in public, as a defender of British sovereignty against European Union powergrabbing in 2014 when he said of then-presidential candidate Juncker that he was the "wrong person" to lead the Commission, saying the Luxembourgish politician had been "at the heart of the project to increase the power of Brussels and reduce the power of nation states for his entire working life".
Juncker said in a speech Monday to the Saarland state legislature in Saarbruecken, Germany: "We now know what the British Parliament doesn't want, but we haven't heard what it wants".
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