The BBC reported that Johnson said in his resignation letter that May has put off decisions about how Britain will leave the European Union, warning that "we appear to be heading for a semi-Brexit, with large parts of the economy still locked in the EU system, but with no United Kingdom control over that system".
Attorney General Jeremy Wright, meanwhile, will be taking up the post of Culture Secretary.
Hunt, the MP for South West Surrey, has served as Health Secretary since 2012.
Eight Cabinet ministers have now departed Mrs May's team over the past year.
Boris Johnson resigned following a day of speculation on Monday afternoon. "I would be surprised if one [a leadership contest] is precipitated and if there is, I suspect she'd win it", he told LBC Radio.More news: Microsoft's Entry Level Tablet Arrives in August for $399 — Surface Go
Justice Secretary David Gauke insisted the Cabinet was now fully behind Mrs May's approach on Brexit after the departures.
How could Theresa May be ousted as Tory leader?
Under Conservative Party rules, a confidence vote in a leader can be triggered if 48 lawmakers from May's Conservative Party request one.
Conservative MPs will then hold a series of ballots to whittle the list of contenders down to two, with the last place candidate dropping out in each round. Many took the opportunity to make fun of the former Brexit Secretary over his short tenure and the timing of his decision to quit.
He was widely reported to have told the Chequers meeting on Friday that putting a positive gloss on Mrs May's Brexit package would be like "polishing a turd", though he later joined other ministers in signing up to the package.More news: Roger Federer cruises into Wimbledon quarter-finals
"In a resignation letter, Davis said the "'common rule book' policy 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".
The two resignations have left May badly exposed and raised questions over whether she will stand firm in her commitment to pursuing a "business-friendly" Brexit, or will be faced with more resignations and calls to quit herself.
Lawmakers howled as a beleaguered May paid tribute to two senior members of her Cabinet who resigned rather than accept her proposals.
The PM said that after months of offering ministers "considerable latitude to express their individual views" she is now imposing order on her Cabinet who must back the Government line. "It should be a chance to do things differently", adding "that dream is dying, suffocated by needless self-doubt".
In 2016, following the resignation of Prime Minister David Cameron, Johnson was widely considered a front-runner in the race that May eventually won, but he ruled himself out after the defection of a key ally.More news: IOS 11.4.1 brings USB Restricted Mode but it has a bug
Mr Johnson even proposed a toast to the deal - and suggested writing a joint newspaper column with Chancellor Philip Hammond. Davis slammed the agreement, which promotes a "soft exit" from the European Union versus a swift, clean break, as giving away "too much away, too easily" to E.U. negotiators.
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