The Sunday Times claimed pro-Brexit lawmakers are prepared to topple May if she tries to compromise and keep Britain in the EU's tariff-free customs union, an option favoured by her Treasury chief Philip Hammond.
Speaking in Shanghai on the final day of Theresa May's trade mission to China, Dr Fox said remaining in a customs union with the EU after Brexit would prevent Britain pursuing other trading opportunities around the world.
But within minutes of her making the statement on television Housing Minister Dominic Raab popped up on another channel where he ruled out staying in any form of customs union.
The backbencher accused ministers of being "vague" and "divided", saying Ms May "sticks to one policy [while] the Chancellor keeps advocating another".
Reports last week had suggested Downing Street was willing to remain in a union with the EU for goods, but not services.
Home Secretary Amber Rudd said Britain was open to "either a customs arrangement or a customs partnership" as part of a deal to deliver "frictionless" trade.
While this theoretically leaves open the ability to strike independent trade deals with the rest of the world, particularly for services, there would be major limits to Britain's room for manoeuvre.
The accusations of cabinet divisions and disunity have kept coming from Brexit-supporting MPs over the past week.
"Now we're not going to surrender before we have that battle", Rudd told the BBC when asked whether ministers were pursuing what the European Union has already ruled out as "having its cake and eating it" - demanding tariff-free access to the EU's market while controlling immigration. Or does she ignore the warnings of business and put cross-border trade with our biggest market at risk in pursuit of so far unproven gains to be made in Asian and USA markets? And what about the customs union, another thorny issue that key ministers have very different views on?More news: Tourism Australia's 'Dundee' Ad Finally Bloody Aired During The Super Bowl
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"Her words come at the beginning of a week when the government will flesh out some of its demands in negotiations with the European Union on their future relationship and Brexit negotiations start in earnest again".
'That is the deal we are looking for.
"Her MPs will back her, because we are overwhelmingly at one with the majority of the British people who now want a clean Brexit and end to the present uncertainty".
"It is time for all her ministers to back her too and to end the confusion they are fomenting in Government".
Andrew Turnbull, who led the civil service under Prime Minister Tony Blair, has accused hardline Brexiteers of using the tactics of "pre-emptive scapegoating" against government officials which he said remind him of what right-wing German nationalists were doing in the run up to WWII, the Guardian reported.
She also affirmed minister to be unified over Brexit at a level a lot higher than critics believe.
In a warning to her Cabinet colleagues insisting the United Kingdom leave the customs union, she added: "If you set up trade barriers, there is a outcome to a trading nation".
"We need to have this wider agreement".
A senior government source says the prime minister wants to "put this to rest".