Britain will have a “great deal of freedom” to negotiate a trade deal with the US under Theresa May’s Brexit plan, the international trade secretary has said.

Following the prime minister’s meeting with US president Donald Trump at Chequers, Liam Fox said “very positive” discussions had been taking place with US officials on a future deal after Britain has left the EU.

In a joint press conference with May on Friday, Trump appeared to row back on an earlier newspaper interview in which he said the government’s latest proposals would kill off the prospect of a deal with the US.

Fox said that ministers had been able to explain details of their plan – which would see Britain maintain a “common rule book” with the EU covering standards for trade in goods – 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. At the same time, he stressed that Britain would not admit US agricultural products that did not meet current standards.

“We will have complete freedom in terms of market access of how much of those approved goods come into the United Kingdom, so if we want to reduce the tariff, for example, on American cars we would have the freedom to do so,” he told the BBC Radio 4 Today programme.

“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 European Union would.

“The standards would be the same. That therefore removes the need for inspection at the border. But in terms of market access, the United Kingdom would have a great deal of freedom.”

Fox said that unlike fellow Brexiters Boris Johnson and David Davis, he had not considered resigning over the compromise plan hammered out by senior ministers and he urged the EU to take up the proposed offer in the interests of all their people.

“We are making a realistic and positive offer to the European Union. I hope we will get their support because what we need to have is a people’s Brexit, not a bureaucrat’s Brexit,” he said.

“We now need to think about the wellbeing of people across Europe, about their prosperity, their jobs, their security, and not about the abstractions of the bureaucrats of Brussels.

“I think most people in Britain will think this is that this is a fair, pragmatic and reasonable approach.”

However, the former permanent secretary at the Department for International Trade, Sir Martin Donnelly, warned the UK would have to give more ground if it wanted a deal with Brussels.

“If we want to have a lot of scope to do trade deals around the world, it is going to be very difficult to have a close relationship with the European Union,” he told the Today programme. “That’s the key trade-off that we have got to deal with.

“It is going to come down, I think, to how far the UK is prepared to follow the same rules and take on the same obligations as others in the European Union to retain access to that market. If we don’t want to do that, clearly we have got more scope to do trade deals internationally.”

On Friday, Trump said he had been convinced that May’s Brexit white paper did not preclude a US trade deal, contrary to his overnight interview with the Sun.

He said that he had apologised to May for the article over breakfast and that she had told him, “don’t worry, it’s only the press” – a revelation of a private conversation that drew a grimace from the prime minister.

According to his former adviser, Steve Bannon, Trump “gave her some pointers and pulled her off to the side and said: ‘Hey, if I was doing this, here is how tough you have got to be because these guys are not going to let you go.’”