Outside of the EU, Britain falls under the rules of what's called the "World Trade Organisation", who basically govern trade between countries who don't have special deals.
The Government has admitted its proposals are unprecedented, but Dr Fox said Mr Barnier's response "makes the chance of no deal greater".
Britain's global trade minister says it's likely the United Kingdom will fail to agree upon a divorce deal with the European Union before it leaves the bloc in March the latest in a growing chorus of warnings that the negotiations are in crisis.
"I think the intransigence of the commission is pushing us towards no deal", he said.
"But the worldwide trade secretary is right to say there is a risk of the negotiations not succeeding and the government has to prepare for all eventualities".
May's office said in an emailed statement Sunday that while the government is "confident" of securing a deal, "it is the job of a responsible government to prepare for all scenarios".
British Prime Minister Theresa May heads to President Emmanuel Macron's "summer Elysee Palace" on the Mediterranean coast on Friday, seeking to soften resistance to a Brexit plan which has upended her government while failing to win over sceptical European Union negotiators. He added: "It's up to the EU27 to determine whether they want the EU Commission's ideological purity to be maintained at the expense of their real economies". May to have the "backbone and confidence" to back Britain in the negotiations with the EU.
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.
The UK and European Union say they want agreement before the exit on 29 March 2019.