Mr Hunt held a Q&A on Twitter last night, when the two contenders to be Prime Minister had been scheduled to debate with one another on Sky News – Mr Johnson refused to appear.
It came as the favourite gave his first broadcast interviews of the contest, refusing to respond to questions about his private life following a row with his partner that saw neighbours call police to her flat.
Mr Johnson would not comment on suggestions that his campaign had released a staged photo of him sitting with Carrie Symonds, his girlfriend, following the incident in the early hours of Friday morning.
And in a bizarre, rambling response to a question about what he does to unwind, the former foreign secretary said he likes to “paint model buses” made out of old wine boxes.
It prompted one MP supporting Mr Johnson to say there was “something not right” with his campaign.
“He’s not firing on all cylinders,” they told The Scotsman. “It feels like the last three weeks of the  general election, when we realised, hang on, this isn’t going well.”
Mr Johnson confirmed he would take the UK out of the EU without a deal, saying he would go ahead with Brexit on 31 October, “do or die, come what may” – and he challenged Mr Hunt to make the same promise.
In an interview with the BBC, Mr Hunt said he would never mount a personal attack on his rival, but added: “We had a lot of discussion about ‘how’, we need to have a discussion about ‘who’.
“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.”
The Foreign Secretary added that delivering Brexit was “about the personality of our PM”.
“If you choose someone where there’s no trust, there’s going to be no negotiation, no deal. And quite possibly a general election, which could mean we have no Brexit either,” he said.
Mr Johnson had kept out of the public eye since the news broke on Friday about the row at the south London home he shares with Ms Symonds.Police were called by worried neighbours after his partner was heard screaming and shouting “get off me”.
His campaign attempted to step up a gear with a series of media appearances and events, but the approach appeared to backfire in his third broadcast interview in less than 24 hours, when Mr Johnson claimed he made models of buses out of old wine crates to relax.
The former mayor of London, whose term in office included the introduction of the new Routemaster bus, told TalkRadio: “I like to paint or I make things… I get old wooden crates, right? And I paint them. I suppose it’s a box that’s been used to contain two wine bottles. It will have a dividing thing, and I turn it into a bus and I put passengers… I paint the passengers enjoying themselves on the wonderful bus.”
On Monday night, Mr Johnson told the BBC: “I do not talk about stuff involving my family, my loved ones.
“And there’s a very good reason for that. That is that, if you do, you drag them into things that… in a way that is not fair on them.”
And speaking to LBC Radio yesterday, Mr Johnson refused to comment when repeatedly challenged over whether his campaign was behind the release of a picture of him with Ms Symonds, in an attempt to show their relationship had not broken down.
Asked where the photograph had come from, Mr Johnson said: “The longer we spend on things extraneous to what I want to do… the bigger the waste of time.”
In testy exchanges, he said there are “all sorts of pictures of me out on the internet which pop up from time to time”. When host Nick Ferrari suggested his hairstyle indicated it was an old picture, he said: “This conversation is now descending into farce.”