Home Secretary Amber Rudd has informed May and other government officials that the possible exodus could take place unless the ministers are allowed to support an amendment in the House of Commons that extend the duration of the Article 50 of EU’s Lisbon Treaty beyond the original March 29 deadline for Brexit.
If approved, the new plan prevents May from proceeding with her previous threats of a no-deal divorce, which analysts say would have unpredictable ramifications.
Tabled by Labour’s Yvette Cooper and former Tory ministers Oliver Letwin and Nick Boles, the new amendment would come up for debate and vote on January 29.
Rudd has asked May to allow a free vote on the bill to prevent a mass walkout from 10 Downing Street.
“Amber is telling Downing Street to make it a free vote on behalf of lots of people,” a Member of Parliament (MP) told The Times.
May's cabinet is deeply divided on Brexit with pro-Europe ministers like Rudd, Chancellor of the Exchequer Philip Hammond and business secretary Greg Clark calling on her to follow a softer strategy.
The no-deal option is also faced with strong opposition in Parliament.
Despite voting down a deal she had already clinched with the EU, lawmakers have made it clear that they would not allow a no-deal exit either.
May lays out plans for new deal
In response to last week’s embarrassing defeat of her deal in the House of Commons, May returned to the chamber on Monday with her proposals for a new deal.
She said she had identified three points in the previous deal that needed to change, including the engagement of Parliament in EU negotiations, stronger protection for workers’ rights and ensuring commitment to “no hard border” in Northern Ireland and Ireland.
The previous deal’s failure was largely blamed on the so-called backstop clause, which allows the EU to include Northern Ireland in its customs union and move the border to the Irish Sea until a mechanism is found for bilateral trade.
The deal's opponents say the clause undermines the UK’s sovereignty because it effectively separates Northern Ireland from mainland Britain.
May’s disorderly handling of Brexit has already resulted in a wave of resignations.