Sir Graham refused to say whether he had received any such letters.
Both Rookie MP Ben Bradley, who only won a seat in the House of Commons previous year, and Maria Caulfield, elected in 2015, resigned as vice chairs of May's governing Conservative Party, citing disagreement with the soft-Brexit announced at a meeting Friday of her full cabinet.
The conflicting accounts came after Mr Davis resigned dramatically at midnight last Sunday, followed by Mr Johnson 15 hours later.
"I would like to thank you warmly for everything you have done over the past two years as Secretary of State to shape our departure from the European Union", she said.
In a sign of the level of concern in Number 10, Mr Smith had written to MPs in an apparent attempt to gag backbenchers from speaking out against the proposals in the media. She thanked him for his work.
The plan was unveiled amid shambolic scenes in the House of Commons, as Speaker John Bercow suspended proceedings in the middle of a statement by new Brexit Secretary Dominic Raab so MPs could go and get a copy of the White Paper he was announcing.
Former Prime Minister Tony Blair has repeatedly called for reversing Brexit, echoing other critics such as French President Emmanuel Macron and billionaire investor George Soros, who have suggested that Britain could still change its mind.
Chief whip Julian Smith was inviting MPs to Downing Street to hear from the Prime Minister's chief of staff Gavin Barwell for a briefing on the Chequers proposals in a bid to win them round ahead of Monday's meeting.
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.
Unlike Johnson, who was one of the most high-profile Brexit campaigners, Hunt backed the "Remain" campaign during the 2016 referendum.
Labour deputy leader Tom Watson said: "Theresa May's Government is in meltdown".
Giving his views on Mr Johnson's decision to leave the Government, Mr Davis told LBC: "I had to resign because this was central to my job and if we continue with this policy and I was still there, I'd have to present it to parliament and the House of Commons, I'd have to present it in Europe".
The Mail on Sunday has established that Mr Davis phoned several Ministers, including Mr Johnson, on Sunday to tell him he meant to quit.
Facing a deeply divided government, party and country, May sought to find a compromise between Brexiteers and Remainers, who fear the economy will be damaged by cutting too many ties with the world's biggest trading bloc. "We are supporting Theresa for PM".
She thought she had done enough to move on with that fraught process at the meeting at her Chequers country residence.
Shadow foreign secretary Emily Thornberry tweeted: "Government resignations since autumn: 1 November 2017 - Fallon 8 November 2017 - Patel 20 December 2017 - Green 29 April 2018 - Rudd 8 January 2018 - Greening 8 July 2018 - Davis There have been six resignations in 249 days".
Creating a political storm when he compared European Union efforts to create a superstate to Hitler's attempt to rule the continent, saying: "Napoleon, Hitler, various people tried this out, and it ends tragically".
Asked by SNP MP Pete Wishart if she anticipated a no-confidence vote, Mrs May replied in the Commons: "Nice try but I'm getting on with the job of delivering what the British people want".
The Liberal Democrats, who back a second referendum on Brexit, said on Twitter: "The resignation of David Davis is yet more evidence of the chaos of this Tory Brexit".