Among Tory Brexiteers there was deepening anger at the proposals agreed at Friday's meeting of the Cabinet at Chequers which they branded as "Brexit in name only".
With 316 Conservative MPs in the House of Commons, Sir Graham must receive 48 letters to call a ballot.
Besides their senior colleagues, Brexit junior minister Steve Baker and two ministerial aides also resigned from the government on Sunday.
Allies of Mrs May said that just six MPs expressed dissent in the course of the meeting.
"I'd be interested to see the actual white paper because there may be some nuances in it which may make people feel a bit happier", she told Sky News. He also said he will "step up the preparations for "no deal" in case the spirit of pragmatism and compromise that we have extended is not reciprocated".
The Suddeutsche Zeitung says Davis and Johnson "ran away" from May, who it says is now in the "most hard crisis" of her time in Downing Street.
The Prime Minister said she was "sorry and a little surprised" over Johnson's resignation, after "productive discussions" on Friday.
While Johnson was one of the most high-profile Brexit campaigners, Hunt backed "Remain" during the 2016 referendum campaign.
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In his resignation letter to May, Johnson accused her waving "white flags" of surrender in negotiations with European Union leaders. "The Prime Minister thanks Boris for his work".
New Brexit secretary Dominic Raab also vowed to strike the "best deal with our European Union friends and partners" while preparing for an increasingly likely no-deal scenario.
There is added pressure for a swift appointment because the foreign secretary is due to join Mrs May at the North Atlantic Treaty Organisation summit in Brussels on Wednesday and take part in the events of US President Donald Trump's visit to the United Kingdom, which begins on Thursday.
Reflecting on Mr Hunt's time at the DH, NHS Confederation chief executive Niall Dickson said: "He inherited a tangled mess of a reorganisation from his predecessor and he had to live with a succession of austere funding settlements which were never going to meet rising demand".
Mr Juncker said: "This clearly proves that at Chequers there was clearly a unity of views in the British Cabinet".
He added, "This is a time when the world is looking at us as a country, wondering what type of country we are going to be in a post-Brexit world".