Hunt's comment comes as Johnson, the former Foreign Secretary, faces pressure to answer questions about a row with his partner in the early hours of June 21 which led to police being called to his London home, the BBC reported.
Writing for the Sunday Times, Hunt called for a "fair and open contest, not one that one side is trying to rig to avoid scrutiny".
Following a vote on June 20, Hunt and Johnson will now go head-to-head to become the country's new Prime Minister. The vote resulted in the elimination of Home Secretary Sajid Javid and Environment Secretary Michael Gove.
Hunt in the Sunday Times accused Johnson of refusing to take part in TV debates and avoiding media interviews.
He said he was "not interested" in Johnson's private life but told him to "man up" and debate with him 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."
Hunt also said that he wanted to "quiz" Johnson on how he could guarantee the UK would leave the European Union (EU) on October 31 if Parliament voted to stop a no-deal Brexit, as it did in a non-binding vote in March.
After outgoing 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 October 31, the BBC reported.
Next month around 160,000 Conservative members will choose the next leader of the Party - and the next Prime Minister.
Members will receive their ballots between July 6 and 8, with the new leader expected to be announced in the week beginning July 22.
( With inputs from IANS )