According to UK media reports, Conservative party MPs in favour of a complete break from the EU have threatened to submit a "tsunami" of letters demanding a vote of no confidence unless May backs their demand of a complete exit from the EU customs union.
"We do not wish to be in 'a' customs union, 'the' customs union or 'an' customs union, said Jacob Rees-Mogg, among the most outspoken pro-Brexit Tory MPs and tipped for a key Cabinet role in the event of May's forced exit.
However, Brexiteers insist they want to be able to do free trade deals on goods as well as services with the rest of the world.
According to 'The Sunday Times', UK foreign secretary Boris Johnson has been approached to lead a coup by creating a team of so-called "Three Brexiteers" with him as Prime Minister, environment secretary Michael Gove as his deputy and Rees-Mogg as Chancellor.
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Senior figures inside the Tory party have indicated that a poor performance in local elections scheduled in London for May this year could trigger a definitive no-confidence vote.
A formal vote of no confidence in May can be held if Sir Graham Brady, the chairman of the Tory backbench 1922 Committee, receives letters from 48 MPs demanding a contest.
He alone knows exactly how many letters have been submitted so far.
Meanwhile, another senior Conservative Party MP and staunch Brexiteer, Bernard Jenkin, wrote in 'The Sunday Telegraph' that the Cabinet was "vague and divided" over Brexit and must get its act together.
UK Chancellor Philip Hammond had recently suggested that the UKs relationship with the EU would change "very modestly" after Brexit. A stand that was roundly criticised by those in favour of a hard Brexit and a complete break from the economic bloc.
May will chair two meetings of the Cabinets Brexit sub- committee on Wednesday and Thursday, hoping to reach an accord and agreement between her warring ministers about the nature of the future trade deal Britain will seek from Brussels. She faces a tough task balancing both sides of the hard Brexit divide and asserting her authority to lead the government.