Jeremy Hunt has called on Boris Johnson not to be a "coward" by avoiding a live TV debate with him on Sky News this week.
Stepping up pressure on his rival, the foreign secretary insisted that while he has no interest in discussing Mr Johnson's private life, he wants to challenge him on television over his commitment to taking the UK out of the EU by 31 October.
In an article in The Times, Mr Hunt said: "A new prime minister needs the legitimacy of having made his arguments publicly and having them subjected to scrutiny.
"Only then can you walk through the front door of No 10 with your head held high instead of slinking through the back door, which is what Boris appears to want."
He added: "Don't be a coward Boris, man up and show the nation you can cope with the intense scrutiny the most difficult job in the country will involve."
Mr Hunt has maintained his commitment to a Conservative leadership hustings to be hosted by Sky News tomorrow night, and called on his opponent to confirm his own attendance or risk disrespecting the party's members.
He told Sky News: "It's important that in the next two weeks which are the only period in which party members can see us head to head before they get their ballots, that they hear from us.
"My worry is that Boris has got a coalition of people like Matt Hancock who wants no deal off the table and Mark Francois who wants no deal - that coalition is going to dissolve quickly when you have to make decisions.
"In politics you can sometimes fudge, but on Brexit you can't."
Mr Hunt urged Sky News to continue with the debate on Tuesday evening, leaving an empty chair if Mr Johnson doesn't turn up.
The foreign secretary also pointed to a claim made by Tobias Ellwood that there are about a dozen MPs prepared to "bring down a Johnson government" to avoid no-deal, and said a general election would be disastrous.
On Sunday, Mr Hunt told Sky News that he thought any candidate for prime minister needs to show he can answer "difficult questions".
He said: "What Boris needs to do is to engage properly in this leadership debate, not to shy away from the Sky News live debate that's scheduled for Tuesday evening, which he's been invited to [and] I've said I'm willing to go to.
"This is an audition to be prime minister of the United Kingdom, and Boris needs to show that he's prepared to answer difficult questions."
Mr Johnson appeared to deliver a retort to his rival to be prime minister as he promised "we are not going to bottle it" on the EU exit date of 31 October - deal or no-deal.
Mr Johnson remains under pressure to explain why police were called to the home he shares with partner Carrie Symonds, but in his column on Monday for the Daily Telegraph, he attempted to switch attention back to political rather than personal issues.
He wrote: "We must leave the EU on Oct 31 come what may. It will honour the referendum result, it will focus the minds of EU negotiators.
"It is absolutely vital that we keep our eyes on the prize. It has been a long and parching march - but the oasis is finally in sight.
"We are just over four months away from the date on which, by law, we must leave the EU; and this time we are not going to bottle it. We are not going to fail.
"This time we are not going to shrink in fear from the exit, as we have on the last two occasions."
Mr Johnson's campaign was rocked by revelations that police officers were called to the south London flat he shares with Ms Symonds by a neighbour who claimed to have been "frightened and concerned" after hearing shouting, "a loud scream" and banging coming from the property.
The police attended and found everyone was safe, with no offence having occurred.
When asked at hustings in Birmingham on Saturday about the incident, Mr Johnson said people "are entitled to ask about me and my determination, my character and what I want to do for the country", but he would not elaborate on the police visit.
Despite increasing calls to explain what happened, Mr Johnson has steadfastly refused to comment.
A poll carried out after the reports emerged suggested Mr Johnson's support among Tory voters had dropped by more than half while among the general electorate it indicated he had slipped into second place behind Mr Hunt.
Questioned about Mr Johnson's apparent row with his partner on Friday, Mr Hunt told Sky News this morning: "We all have things that we would not want in the public arena.
"It would be an easy hit [to criticise Johnson] but it would be the wrong hit."
He added his belief that the public "don't want leadership candidates throwing brickbats at each other" but that it is "disrespectful" of Mr Johnson not to appear in head to head debates.
Priti Patel, an MP in Mr Johnson's team, told Sky News it was a police matter but it was now "done and dusted".
She denied that Mr Johnson had given insufficient information about his plans, saying he was out speaking to members, and had already done one debate and a hustings.
On Saturday, the neighbour who rang the police went public after suggestions that his recording of the row had been leaked to the Guardian with political intent.
Tom Penn said the allegations were "bizarre and fictitious", explaining in a statement to the paper that he dialled 999 after hearing shouting coming from his neighbour's flat.
Mr Johnson and Mr Hunt are expected to take part in a head-to-head ITV debate on 9 July hosted by Julie Etchingham.
Sky News is also due to host a debate on Tuesday, moderated by Kay Burley, which Mr Hunt has agreed to take part in - but Mr Johnson has so far refused to commit.