Sky News has said it will be forced to cancel a debate between the two men vying to be the next Tory leader unless Boris Johnson agrees to take part.
The broadcaster hoped to be the first to stage a head-to-head debate between Mr Johnson and his rival Jeremy Hunt.
But it said Mr Johnson had "so far declined" its invitation and the event would not go ahead without him.
Mr Hunt said it was "incredibly disappointing" and risked "cheating" the country out of a "proper contest".
He said the next prime minister would not have the popular mandate to take difficult decisions on Brexit and other issues if they were not prepared to face scrutiny now.
Mr Johnson has faced three days of questions over his private life after a row with his partner Carrie Symonds.
The former foreign secretary has declined to comment on the nature of the argument in Ms Symonds' London home, which led to the police being called early on Friday morning.
He has also been accused of avoiding media scrutiny more generally, particularly on his Brexit policy.
Allies of Mr Johnson have stepped up their criticism of Ms Symonds' neighbours, Tom Penn and Eve Leigh, for recording part of the argument and sharing it with the Guardian newspaper, suggesting their actions were "politically motivated".
Jacob Rees-Mogg, the leader of the European Research Group of Brexiteers, described the couple as "Corbynista curtain twitchers".
The couple have insisted they were motivated solely by concerns for the welfare of Ms Symonds after reportedly hearing raised voices and plates and glasses being smashed.
Mr Johnson remains the frontrunner in the contest to succeed Theresa May, with 160,000 or so Conservative Party members choosing their next leader by the end of July.
Mr Hunt has challenged him to a series of live TV debates over the next 10 days.
But Sky's planned debate on Tuesday now looks unlikely to go ahead.
"Jeremy Hunt has agreed to take part, but Boris Johnson has so far declined the invitation," the broadcaster said in a statement.
"We stand ready to host a debate tomorrow evening if both candidates make themselves available," it said. "Without both candidates, tomorrow's debate will not take place."
Mr Johnson has agreed to take part in a one-on-one debate with Mr Hunt on ITV on 9 July, but by then voting papers will already have been sent to party members.
The two men are also taking part in a series of hustings for Tory members across the UK.
Mr Johnson and Mr Hunt took part in a five-way debate earlier in the leadership contest on the BBC last week, but Mr Johnson refused to join a similar event on Channel 4.
Writing in the Times on Monday, Mr Hunt urged his rival not to be "a coward" and to take part in the Sky debate. He insisted, though, he was "not interested" in Mr Johnson's private life.
He said Mr Johnson must explain how he could guarantee the UK would leave the EU on 31 October if Parliament voted to stop a no-deal Brexit, as it did in a non-binding vote in March.
Analysis: By Norman Smith, BBC assistant political editor
The response from Team Hunt is to call him "bottler Boris" - to accuse his campaign of being complacent and not trusting their man to turn up.
Team Hunt will also be pretty annoyed because they wanted this opportunity to boost their man's profile and get him some name recognition.
There's no getting away from the fact that he's just not as well-known as his rival.
For their part, Mr Johnson's supporters say he's taking part in 16 hustings events across the country and that is scrutiny enough.
Responding to the Sky announcement, Mr Hunt suggested his rival was behaving like Theresa May during the 2017 general election, when she avoided TV debates and even sent another minister in her place.
"Frankly, the Conservative Party will start to feel cheated of having a proper leadership campaign if the frontrunner isn't prepared to subject himself to the scrutiny of TV debates," he said.
"There's only two weeks now before postal ballots arrive, and we owe the Conservative Party, we owe the country, a proper contest."
In his own newspaper column on Monday in the Daily Telegraph, Mr Johnson said the UK would face a "democratic explosion" if it did not leave the EU by 31 October.
"This time we are not going to bottle it. We are not going to fail," he wrote.
His team insist he is facing scrutiny via a series of leadership hustings in front of Conservative Party members taking place over the coming weeks. All of them will be live-streamed for public viewing.