With less than nine months left until Britain is due to leave the bloc, May is sticking to her plan for a "business friendly" Brexit.
The Foreign Secretary quit his role in government hours after Mr Davis stood down as Brexit Secretary.
The Conservative leader is at risk of suffering more damaging resignations with the publication of her White Paper today, which sets out the new Brexit policy in detail.
The initial euphoria over British prime minister Theresa May's cabinet reaching a "collective" agreement for her plans to leave the European Union seems to have been shortlived as the government grapples with departures.
The EU's chief negotiator, Michel Barnier, has insisted the proposals must be workable.
The Prime Minister was greeted by loud cheers from Tory MPs and shouts of "resign" from the opposition benches as she arrived to deliver a statement in which she said her proposals would deliver "a Brexit that is in our national interest... the right Brexit deal for Britain".
Sky News has been told that Brexiteers have "plenty more in the locker" as they seek to "inflict as much damage as possible" on Mrs May and try to force her to change course.
Asked what he would do if the Brexit plan ended up limiting Britain's ability to conduct trade deals with non-EU countries, Fox said: "I don't believe it does". "He s been very, very nice to me, very supportive".
Fallon warned Conservative rebels that a challenge to May's leadership is "the last thing we need".
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"It is the ultimate betrayal of our democracy and people's belief in it". Euroskeptic Conservative lawmaker Peter Bone said party activists felt "betrayed" by the government plan.
Andrew Bridgen became the first Conservative MP to publicly confirm he had sent a letter calling for a vote of confidence in the prime minister, it was reported.
There is, however, a major potential new barrier: The Brexit plan published by the United Kingdom government on Thursday is created to keep the country closely aligned in key areas with the European Union, severely limiting its flexibility to negotiate new trade agreements.
Two years after Britain voted 52 percent to 48 percent to leave the European Union, May is trying to find a middle way between two starkly differing views - within her party and the country - of the U.K.'s relationship with Europe. But within 48 hours of their publication, Davis had quit, saying she had already given too much away to the European Union, and Johnson followed.
Boris Johnson has launched a scathing attack on Theresa May's Brexit strategy, saying the "dream is dying, suffocated by needless self-doubt".
If the European Union shuns the paper, however, sterling could fall below $1.30 as risks of a "no deal" Brexit pick-up, said ING FX strategist Viraj Patel.
The Sun, Britain's biggest-selling newspaper, which has long argued for Brexit, said May had made "blunder after blunder".
Labour leader Jeremy Corbyn said Mr Johnson and Mr Davis had abandoned a "sinking ship", shattering the "illusion of unity" initially surrounding the Chequers plan.
"If people don't like this proposal, what is their alternative?" They've said it must be one or the other, and that a deep and frictionless free-trade arrangement must be bundled with the single market's free movement of people.