Backing Mrs May's proposals could affect his support among Eurosceptic backbenchers.
But for a brief time, it appeared that Theresa May had finally silenced the bickering and stopped the backstabbing over Brexit on July 6 following a day-long cabinet meeting at her country residence, Chequers.
The Chequers plan involves the creation of a "UK-EU free trade area" for goods, governed by a "common rulebook".
Johnson - one of the country's leading Brexit advocates - stepped down on Monday after the resignations of Secretary of State for Exiting the European Union David Davis and deputy secretary Steve Baker.
"The general direction of policy will leave us in at best a weak negotiating position, and possibly an inescapable one", he said.
The "common rulebook" plan "hands control of large swathes of our economy to the European Union and is certainly not returning control of our laws in any real sense", he wrote to Mrs May.
Just over one in five - 22% - think the likeliest outcome is that Britain remains in the European Union, while 7% say they don't know.
Europe's chief negotiator Michel Barnier said Tuesday talks will resume next Monday as scheduled with the new Brexit minister, while refusing to comment on the political infighting consuming London.
She said the deal agreed by the Cabinet after their "productive discussions" would "honour the result of the referendum" and allow the United Kingdom to "take back control of our borders, our law and our money".
Under the Conservative Party's rules, "a confidence vote can not be repeated for another 12 months, so if she survived one, she would have the space to complete the Brexit negotiations", he said.
Arch-Brexiteer Jacob Rees-Mogg, leader of the powerful European Research Group faction within the Tory ranks, said his resignation should force Mrs May to reconsider her approach.
Asked if Brexiteers needed to put the PM's future to a vote of the Conservative Party, he replied "it may well come to that".
Hunt said it was a "massive wrench" for him to leave the department, adding: "I know some staff haven't found me the easiest Health Sec".
Davis himself said it would be "wrong" for his departure to trigger a major party rebellion against May.
Letters calling for a leadership contest have reportedly been submitted to the backbench 1922 Committee by some Conservatives over the weekend.
She paid tribute to Mr Johnson's "passion" in championing a global Britain after Brexit and Mr Davis' work in steering through key Brexit legislation. Brexit-supporting lawmakers were angered by the proposals, saying they would keep Britain tethered to the bloc and unable to change its rules to strike new trade deals around the world.
Environment Secretary Gove's reputation was left "in tatters" after running for the leadership against his supposed ally Boris Johnson in 2016, says The Guardian, and he has kept a relatively low profile in recent months.
Mr Walker campaigned for Remain in the referendum.
"I think it helps her".
Will Boris Johnson sing a new tune now that he is just another unhappy Conservative MP over Brexit?
At a press conference with German Chancellor Angela Merkel in London later Tuesday, May vowed to deliver a "smooth and orderly Brexit that protects jobs, livelihoods and also meets our commitment for no hard border between Northern Ireland and Ireland".
The Prime Minister is expected to appoint a replacement for Mr Davis on Monday.
Theresa May raised the prospect of a Jeremy Corbyn-led government to appeal for Tory unity on Brexit.