Sky News invited Johnson and Hunt to a debate on Tuesday, but cancelled the session because Johnson refused to take part.
"But we will reissue our invitation for Mr Hunt and Mr Johnson to debate live on Sky News next Monday, 1 July", the broadcaster added in a statement. The image is believed to have been released to the press by his campaign team, but speculation over when it was taken led to a defensive response from Mr Johnson.
Talks are "not going to be easy" but "it's not impossible either", because a deal is in both sides' interests, Hunt said.
However, Boris Johnson's campaign has been dented by his personal life.
But it is the party's 160,000 members who will have the final say, and their support appeared undimmed during the first day of "hustings" -- internal party debates to decide the new leader -- in Birmingham, central England, on Saturday.
Political rivals questioned Boris Johnson's refusal to comment on the incident that brought officers to his door early Friday, when a neighbor reported hearing shouting, screaming and banging from the home Johnson shares with partner Carrie Symonds.
But since a story broke last week about a noisy row with his girlfriend that prompted a police visit, but no further action, he has come under increasing pressure to break his silence.
"Of course, we can do it", Johnson told the first of 16 leadership hustings events.
A spokesman for Mrs May declined to comment on the row.
His lead over Hunt with Conservative voters had been slashed from 27 percent to 9 percent since Friday, the poll suggested.
He and Hunt are the last two contestants competing to replace May as Conservative leader and Britain's next prime minister.
"He contradicts himself in every single interview every time he says, "I will be the man who takes us out without a deal, ' yet at the same time it will destroy businesses and at the same and 'I know the parliamentary arithmetic would not support me'".
And they want to get it done in a way that allows us to move on which is why I think people are yearning, their yearning for this great Incubus to be pitchforked off the back of British politics.
The Foreign Secretary warned that trying to leave by Halloween at all costs could stop Brexit from happening after Mr Johnson said that it was a "do or die" date to leave the EU. The victor of the contest will replace Theresa May, who stepped down as party leader after failing to secure Parliament's approval for her European Union divorce deal.
"In that situation, you would have to carry on negotiating", Hunt said.
In all the interviews, the former mayor of London sought to turn the conversation back to his wish to bring Britain out of the European Union by October 31 - with or without a deal.
She added: "We need new approach and that means redoing things, redoing the negotiations with the European Union and being much clearer that we are prepared to leave on the 31st October".
Like Mr Hunt, Mr Johnson promised lower taxes if he won the top job.
When asked the naughtiest thing he had ever done, Hunt said: "When I was backpacking through India, I once had a Bhang Lassi - which is a kind of cannabis lassi - that's the naughtiest thing I am prepared to confess to on this programme".
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