After the cabinet meeting, May tweeted that she was "looking ahead to a busy week".
Last month, Hunt said it was inappropriate for businesses like Airbus to issue warnings about moving jobs because of Brexit and that they should instead be getting behind May in her push for a good deal on leaving the EU.
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".
Speaking in the Commons, Mr Cable said: "The very striking things about the comments that resigning members of the government have made is the way in which they are managing to poison their own well".
Addressing the backbench 1922 Committee, the Prime Minister said the alternative to the party coming together could be a left wing Labour administration.
The outgoing Brexit Secretary said the Government had gone further than it should have in the negotiations, and warned it was a "dangerous strategy".
With 316 Conservative MPs in the House of Commons, Sir Graham must receive 48 letters to call a ballot.
He is replaced at Health and Social Care by Culture Secretary Matt Hancock, who will in turn be succeeded by Attorney General Jeremy Wright.
A two-year negotiating window closes in March 2019.
It comes right before Donald Trump's first visit to Britain as U.S. president on Friday, as well as the North Atlantic Treaty Organisation summit in Brussels tomorrow. But, after much speculation, Mr Johnson announced he would not enter the Conservative leadership race when his ally Michael Gove announced he was running. If Labour were to signal it would definitely do the same, that could, in theory, leave May with a more serious problem: a looming veto over the divorce talks leaving the prime minister in an extremely hard position.
The British leader may have stilled the waters over a possible leadership contest, but some Conservative Brexit supporters are still incensed over what they see as her decision to break her promise for a clean break with the EU.
May's office said in a terse statement that the prime minister had accepted Johnson's resignation. "Brussels must not mistakenly conclude, as EU Council chief Donald Tusk hinted, that one more push will destroy Brexit".
Mrs May surprised many by appointing him to the Foreign Office, a position he has frequently used to forge a distinctive position on Brexit, including by setting out his own "red lines" just days before the PM's crucial speech to the Conservative conference a year ago.
He flew out to Washington DC to convince Mr Trump that Iran would "go for a bomb" if he deserted the agreement, with his words falling on deaf ears as the United States later announced it would be pulling out of the deal.
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