He said many MPs saw Mr Johnson's resignation as a "selfish act".
But Mrs May has insisted that her proposals are the only way of protecting the economy and preventing a hard Irish border, while at the same time scrapping free movement and allowing the United Kingdom to strike new trade deals.
She told Sky News: "I'm happy to hear it will be published on Thursday".
The Prime Minister is hoping to win over Tory Brexiteers after a backlash against her plans for leaving the European Union.
Asked if he would join some other lawmakers in calling for May to stand down, he told BBC Radio: "I won't be one of them".
Mr Davis insisted the pair had not co-ordinated their resignations.
Under Conservative party rules, 48 MPs need to send a letter to Graham Brady, chair of the backbench 1922 Committee, calling for a contest.
The Foreign Secretary backed the proposals at Chequers despite claiming that defending the plans during the meeting was like "polishing a turd".
"If the Brexit deal is rejected by Parliament, then we, the people of Britain, should have the democratic right to determine our own future".
In the Commons, Mrs May will acknowledge that there have been "robust views" around the Cabinet table and a "spirited national debate" since the 2016 referendum decision to leave the EU.
In March 2017, some 54% trusted her to get the best Brexit deal possible and just 33% did not trust her. Leading proponents of Brexit called on her to abandon the deal struck with the Cabinet last Friday at Chequers, her country retreat outside London.
The value of the British pound dropped on worldwide markets after Johnson's announcement, having risen after the announcement Friday that a Cabinet deal on Brexit had been agreed.
J.K. Rowling, who has been known to make her feelings about Brexit pretty clear in the past, waded in on Monday.
"I would be surprised if one [a leadership contest] is precipitated and if there is, I suspect she'd win it", he said.
"I like Theresa May, I think she's a good prime minister", Davis said.
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