Following the Prime Minister’s meeting with President Donald Trump at Chequers, Dr Fox said “very positive” discussions had been taking place with US officials on a future deal.
In his joint press conference with Mrs May on Friday, President Trump appeared to row back on an earlier newspaper interview in which he said the Government plan for Brexit would kill off the prospect of a deal with the US.
Dr Fox said ministers had been able to explain details of their plan during their meetings with the president and his officials.
He said the plan would enable the UK to offer much greater access to US goods than the EU was prepared to allow.
But Britain would not admit US agricultural products that did not meet current standards.
“We will have complete freedom in terms of market access on how much of those approved goods come into the UK,” he told the BBC Radio 4 Today programme.
“So if we want to reduce the tariff, for example, on American cars we would have the freedom to do so. That is essentially what we would be able to offer in any trade agreement.
“We would be able to offer much freer market access than the EU. The standards would be the same.
“That therefore removes the need for inspection at the border.
“But in terms of market access the UK would have a great deal of freedom.”
Dr Fox, who is one of Mrs May’s closest allies, said that unlike fellow Brexiteers Boris Johnson and David Davis, he had not considered resigning from the Cabinet over the compromise plan hammered out by senior ministers.
“We are making a realistic and positive offer to the EU. I hope we will get their support because what we need is a people’s Brexit, not a bureaucrats’ Brexit,” he said.
“We need to think about the wellbeing of people across Europe and not about the abstractions of the bureaucrats of Brussels.
“I think most people in Britain will think that this is a fair, pragmatic and reasonable approach.”
Last night former Conservative leader Iain Duncan Smith said: “Mr Trump has been a bit more diplomatic but the situation remains the same – we won’t be able to trade independently with other countries if we are still tied to Brussels.”