David Davis has criticised Theresa May for admitting she would have to make compromises to the EU beyond the Chequers agreement in order to achieve a Brexit deal, and said he could not vote for what has been proposed because it was worse than staying in.

The former Brexit secretary was speaking on the BBC’s The Andrew Marr Show after the prime minister had said in a column for a Sunday newspaper that she would “not be pushed into accepting compromises” on Chequers that are “not in our national interest”.

Davis, who resigned because he could not endorse the Chequers deal, said May’s words amounted to “an incredible open sesame”, arguing the problem with the UK position was that it was “not the last step” for the EU, and Brussels would not accept it.

His comments come as the Conservative party is increasingly split over the Chequers agreement, which proposed the UK should voluntarily submit to a common rulebook with the EU for food and goods, an idea that the party’s rightwing says amounts to “rule-taking” from Brussels.

Davis said he could not vote for a final deal based on Chequers when it comes to parliament for a vote in autumn because it would have been “rather odd for me to resign over something and vote for it”.

He claimed “my old department has been working on a fallback for some time”, saying he believed it was still possible to strike a “free trade-plus” deal based on arrangements the EU has with Canada, South Korea and Australia – although the government has rejected this.

On the Northern Ireland border, Davis said: “I do think – I’m free to talk more freely now than perhaps when I was a minister – I do think we’ve heavily overemphasised the problem on the Northern Ireland border … This is a much more straightforward issue to deal with if we choose to, if we put the political will behind it, we and the Irish Republic, the two together.”

On the same programme, the international trade secretary, Liam Fox, indicated the government may have to make further concessions. He said the EU had to understand “this is a negotiation”, and added: “We have already set out what we think is a reasonable position for the UK to have in our future trading relationship with Europe. We are waiting for the EU to come back to us with their view.”

Davis, meanwhile, stopped short of saying May would have to resign if she could not get her deal, as he hoped to persuade the government to change its position, and said he was not working with Boris Johnson to oust her.

“I don’t know what Boris wants to do,” Davis said, prompting Marr to respond that Johnson wants to be prime minister. Davis replied: “I don’t know what he is planning to do.”

Weekend reports had suggested a member of staff from a company part-owned by Lynton Crosby, a longtime Johnson adviser, had been seconded to work with the pro-Brexit group Change Britain. CTF said its “staff are often consulted because of their campaign expertise and knowledge”, adding that Crosby had been on holiday for five weeks.

In her article for the Sunday Telegraph, May again ruled out a second referendum. She also emphasised that the UK was ready to walk away without a deal if talks with the EU did not conclude satisfactorily.

“For some sectors, there would be real challenges for both the UK and the EU. But we would get through it and go on to thrive,” the prime minister wrote.

“So we will be ready for a no deal if we need to be. And I will not be pushed into accepting compromises on the Chequers proposals that are not in our national interest.”

Fox later appeared to suggest that former Ukip donor Arron Banks should have been allowed to join the Conservative party as long as he “believes what we believe”.

The pro-Brexit trade secretary refused to say whether he believed Banks’s application should be have blocked, arguing instead that the Conservative party “is a broad church” and that it should be open to “people who voted Ukip before”.

He said: “I’m happy to see people in the Conservative party who believe what we believe”.

When pressed shortly after about Bank’s refused membership Fox added: “I don’t comment on any one individual, but people who believe what we believe are welcome in the Conservative party.”

Banks has been leading a campaign to persuade Brexit supporters to join the Tory party ahead of what could be the election of a new leader, in what has been described by some in the party as a campaign of entryism.

However, Banks own membership application was refused the week before last because he had been deemed to have been campaigning against the Conservatives until recently.

Fox said he believed that what was happening in the Conservative party was very different to what had happened in Labour, describing that as leading to “bitter unpleasant bullying”.