His fellow Cabinet colleague and Brexiteer, Andrea Leadsom, is thought to have told May that she "hated" the plan but would give the prime minister her support.
Brexiteer Tory MP Peter Bone supported his decision to quit, saying: 'David Davis has done the right thing, a principled and courageous decision. Well done David Davis for having the principal and guts to resign. We needed Boris to emulate his hero Churchill.
Labour Party chairman Ian Lavery said: "This is absolute chaos and Theresa May has no authority left".
British Prime Minister Theresa May views a vehicle on a production line during a visit to the Jaguar Land Rover factory on September 1, 2016 in Solihull, England. She can not deliver Brexit and our country is at a complete standstill, while the Tories indulge in their leadership tussling. Britain needs a functioning government'.
Other backbenchers have backed the aforementioned Jacob Rees-Mogg, with one describing the Somerset MP as "our Churchill".
Ireland's foreign minister said on Sunday the proposal was a significant move towards a much less disorderly Brexit, though tough negotiations still lie ahead.
Speaking in Cork earlier today, Tanáiste Simon Coveney said what Mrs May had achieved was politically significant and marked the first time in six months where there was a unified position from the British Cabinet.
He added he would "listen to what the Prime Minister has got to say on Monday evening at the 1922 Committee" before deciding what action to take.
"It is not practicable in customs terms to guarantee free goods transit between the European Union and Britain as well as own free trade agreements with third countries, as Britain wants", said Thilo Brodtmann, head of Germany's Mechanical Engineering Industry Association (VDMA).
An analysis of the Chequers deal circulating within the pro-Brexit European Research Group of MPs was damning about the plan, saying it would lead to "a worst-of-all-worlds "black hole" Brexit where the United Kingdom is stuck permanently as a vassal state in the EU's legal and regulatory tar pit". Whether the Prime Minister will be able to get her plans past the Brextremists in her own party let alone Brussels remains to be seen.
Under the proposals, yet to be presented to the European Union, there would be a free trade area for industrial and agricultural goods, based on a "common rule book" and a "combined customs territory".
Tory MPs are being offered one-to-one briefings in Downing Street over the next few days to win them round before publication on Thursday of the white paper, which will spell out the plan in more detail.
The bloc has long warned it will not accept any "cherry-picking" in relation to the single market's four freedoms on goods, services, people and capital.
A source close to the Foreign Secretary said he would not step down as this would play into the hands of European Union commission president Jean Claude Juncker, German chancellor Angela Merkel and European Union chief negotiator Michel Barnier.
He added: 'The EU has never been keen to facilitate a breaking up of an approach toward the single market in terms of keeping all of the elements of the single market intact and consistent, so I think Britain will find it hard to persuade the EU to support the approach they're now proposing.
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