Whitehall sources said they believed there was only a "very small chance" of the government failing to secure a deal, and that preparations were being stepped up in what they described as the unlikely event of that taking place.
Speaking to the Sunday Times, he claimed European Commission's chief Brexit negotiator, Michel Barnier, blasted Mrs May's Chequers plan on the basis that "we have never done it before" and that it "makes the chance of no deal greater".
It was up to the European Union whether it wanted to put "ideological purity" ahead of the real economy, Fox said.
The Prime Minister held talks with French President Emmanuel Macron on Friday, cutting short her holiday to visit his summer retreat.
European Union countries would lose an average of 1.5 percent of their GDP, while Germany is set to lose £61 billion in trade deals.
And ministers including Foreign Secretary Jeremy Hunt and Brexit Secretary Dominic Raab have also engaged in diplomatic activity in Europe in recent days as the Government seeks to deal directly with individual governments in an effort to keep the Chequers plan alive. "Clearly talking about it is very hard to do without spooking people".
Business groups in the City and beyond have been nearly unanimous that a "no deal" Brexit would be harmful.
Adam Marshall, director general of the British Chambers of Commerce, said: "Government needs to show that it's doing everything within the UK's control so that businesses know what they would need to deal with on day one".
However, Fox told the Sunday Times that it was "essential" to have no deal as an option to increase the UK's negotiating leverage.
"That needs to be much higher in people's minds because of course people's livelihoods and the state of the wider economy does depend on having that information".
"No deal would be a catastrophic failure of government, which no government should survive", he said. Michael Ryan, head of the firm's Northern Ireland operation, said spending such a sum to store goods is "not how we can afford to run a business" and is "cash that I don't have". "Parliament has a duty to prevent it".