Jeremy Hunt has urged Tory leadership rival Boris Johnson not to be “a coward” about facing public scrutiny.
He said he was “not interested” in Mr Johnson’s private life but told him to “man up” and debate with him on Sky News this week.
Mr Johnson, writing in the Daily Telegraph, signalled that his focus was on leaving the EU on 31 October.
He suggested that the UK would face a “democratic explosion” if it did not leave by that date.
In his newspaper column, he said: “This time we are not going to bottle it. We are not going to fail.”
It comes as Mr Johnson faces pressure to answer questions about a row with his partner in the early hours of Friday which led to police being called to his London home.
The Metropolitan Police have said they will not be taking any further action on the incident.
Writing in the Times, Mr Hunt called for a “fair and open contest, not one that one side is trying to rig to avoid scrutiny”.
He accused Mr Johnson of refusing to take part in TV debates and avoiding media interviews.
Mr Johnson chose not to appear on Channel 4’s televised leadership debate earlier this month, but did appear on one with the BBC.
Mr Hunt urged his rival to appear on a Sky News debate on Tuesday evening: “I’ll be there. So 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.”
‘Faith in democracy’
The former foreign secretary did challenge Mr Hunt to debate with him on ITV, which he accepted. But Mr Hunt criticised the timing of that event, saying voting papers would have already been sent to party members.
Writing in his weekly newspaper column, Mr Johnson reiterated his commitment to leaving the EU on 31 October.
He said it was “disgraceful” the UK was still in the EU three years after it voted to leave in a referendum.
After Prime Minister Theresa May failed to get her Brexit deal through Parliament earlier this year, the date of the UK’s departure for the EU was moved to 31 October.
“I know that if we fail again, we face a democratic explosion and a deluge in which both major parties may be swept away,” he wrote.
“When we come out of the EU, on October 31… we will see a return of trust in all politicians – the trust that simply disappeared after we failed to leave on March 29.”
Exiting the EU would “renew the national faith in democracy”, Mr Johnson added.
He did not address questions about his private life in the column.
Mr Hunt said he wanted to “quiz” Mr Johnson on 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.
Health Secretary Matt Hancock, who withdrew from the leadership contest after coming sixth in the first ballot of the party’s MPs, told BBC Breakfast Mr Johnson had the “best chance” of securing a new Brexit deal with the EU.
Mr Hancock said it was “total nonsense” to suggest Mr Johnson was not open to scrutiny, drawing attention to the various hustings he has taken part in.
“He’s got the energy, he’s got the support from right across the party, and I think that’s why he’s the right man for the job,” Mr Hancock added.
‘No confidence vote’
In a separate development, defence minister Tobias Ellwood told the BBC’s Panorama programme that “a dozen or so” Conservative MPs would support a vote of no confidence in the government to stop a no-deal Brexit.
“I believe that absolutely is the case. I think a dozen or so members of parliament would be on our side, would be voting against supporting a no deal and that would include ministers as well as backbenchers,” he said.
Next month around 160,000 Conservative Party members will choose the next leader of the Tory Party – and the next prime minister.
Members will receive their ballots between 6 and 8 July, with the new leader expected to be announced in the week beginning 22 July.