Foreign Secretary Boris Johnson backed the proposals at Chequers, despite claiming that defending the plans was like "polishing a turd" during the meeting.
Earlier, British Foreign Minister Boris Johnson, during a closed meeting of Tory supporters and donors, severely criticized British Prime Minister Theresa May.
"We must be open-eyed about the actions of hostile states like Russian Federation who threaten the potential growth of the Eastern Neighbourhood and who try to tear our collective strength apart".
This new broadside comes only hours after Mrs May made it clear that collective cabinet responsibility, suspended during the referendum (and to some extent throughout the Brexit process), was now fully restored.
Insiders say he used colourful language to describe Mrs May's proposals during a marathon Brexit session at Chequers last Friday although he did eventually get behind her blueprint for a soft Brexit.
In a message to the Cabinet, he said: "All those harbouring leadership hopes have done their ambitions fatal harm".
Yet after reflecting on the situation, Mr Davis and Mr Baker apparently decided they could not live with such a soft Brexit after all.More news: Most senior Brexit official quits United Kingdom government
Others have urged Jacob Rees-Mogg to stand, the paper reports.
The Prime Minister will address Tory MPs on Monday and her chief of staff Gavin Barwell has been engaged in efforts to explain the Chequers deal to concerned colleagues alongside Chief Whip Julian Smith.
May unveiled the plan at a closed door government meeting on Friday, siding with those in her divided cabinet who favor closer ties with Europe while ordering those who support a cleaner break to back her policy or quit.
John Longworth, one-time director general of the British Chambers of Commerce and co-chairman of the pro-Brexit group Leave Means Leave, pulled no punches with his message, saying: "Those who campaigned for Leave but have not resigned will see their reputations in tatters".
The Prime Minister was then quick to be the first to get out her version of what had been agreed by giving an interview to the BBC's political editor who had been waiting all day.
Their complaints raise a question mark over whether May can win backing in parliament for her plans if any deal with the European Union is agreed later this year, and some suggest several of them could try to trigger a leadership contest against her.
The document has not been endorsed by the ERG but was "devastating", a Brexiteer source said.More news: America's first dogs came from Siberia, disappeared after Europeans arrived
One Tory MP told Sky News: "We've been mugged".
Full details will be published in a white paper on Thursday, with Brussels expecting a round of negotiations in the week commencing July 16.
An insider who was present at the Chequers summit said: "They all put up a courageous fight but they were outnumbered and worn down during a long, hard day".
He said the briefings "will massively calm the nerves of people who have been basing their views on the speculation that has been floating around in the media and social media".
Mr Gove told the BBC her plan was a compromise.
Although freedom of movement would come to an end, a mobility framework will ensure British and European Union citizens can continue to travel to each other's territories and apply for study and work. Consequently there will be a significant cost to the United Kingdom economy: services accounts for 80pc of the United Kingdom economy and 44pc of the UK's exports - the highest among major economies - and "it is overwhelmingly where its comparative advantage lies (the United Kingdom runs a large trade surplus with the European Union and non-EU in services)", says Daniel Vernazza, chief United Kingdom economist at UniCredit.
"But we'll do it in a way that protects drops and enhances our economy for the future".More news: Starbucks, citing environment, is ditching plastic straws
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