Jeremy Hunt has suggested there is “no trust” in Boris Johnson to fulfil his promises on Brexit, telling the BBC he believes he has the better personality to be prime minister.

Speaking after a war of words with his Conservative leadership rival, whom Hunt branded a coward for turning down a debate with him on Sky News on Tuesday night, Hunt said 31 October was a “fake deadline” and could lead to a snap general election.

He made the comments after Johnson wrote to him asking him to commit to taking the UK out of the EU on 31 October come what may. Hunt said he believed that would be “more likely to trip us into a general election before we’ve delivered Brexit, and that would hand the keys to Jeremy Corbyn and then we’d have no Brexit at all”.

Hunt said he believed a new prime minister would know soon whether a new deal was possible. “If there isn’t and if no deal is still on the table I’ve been very clear. I will leave the European Union without a deal,” he said. “But I’m not going to do that if there’s a prospect of a better deal and if I did it, it would be with a heavy heart, because businesses up and down the country would face a lot of destruction.”

He said he would aim to renegotiate the withdrawal agreement with an alternative arrangement to the Irish backstop, which he said was an approach “not too different to what Boris wants”.

However, he said he believed he had a far greater chance of succeeding in Brussels than Johnson. “Who is the person that we trust to send to Brussels on behalf of the British people and come back with a deal, and that has to be someone that they trust, that they’re prepared to talk to, because in the end you don’t do a deal with someone you don’t trust,” he said.

Hunt said he was not trying to paint his rival as untrustworthy. “I would serve Boris Johnson to the very best of my ability and make his prime ministership a success and I hope he’d do the same for me,” he said. “But obviously members of the Conservative party need to make their choice about who will come back with that deal.”

Hunt said he had been “waiting for this moment for 30 years of my life … I have been sitting around that table thinking about how I want to transform our country.

“I think this is a moment when I look at Brexit and this incredible moment in our history and we could really unleash our potential, and that’s what really gets me up in the morning.”

The foreign secretary was questioned about his view that the legal time limit for an abortion should be reduced from 24 weeks to 12, reiterating he would not seek to change the law.

He said if parliament voted on legal time limits in a private members’ bill, his vote would be “a matter of conscience” but declined to say which way he would vote.

Pressed on whether he would personally vote for it, he said: “Well, I’d have to look at what that bill was but I think for people watching this programme, what they want to know is as prime minister I recognise this is a free vote matter and I wouldn’t seek to change the law.”

On Tuesday night, Hunt took questions on Twitter under the hashtag #BoJoNoShow, a reference to his rival’s refusal to participate in a head-to-head debate that Sky News had been hoping to host that evening.

Hunt’s Twitter responses included an admission that “police cuts went too far” and an expression of regret over the NHS junior doctor strikes that took place in January 2016 while he was health secretary.

He ended the session by posting: