The pair are battling for the votes of an estimated 160,000 Conservative party members, with the winner declared on 23 July, and taking office on 24 July.
Mr Johnson has sold himself as the charismatic leader to guide Britain through troubled times, despite questions over his competence and populist rhetoric.
But after weeks spent avoiding media interviews, he has been accused of ducking scrutiny.
He has been called upon to explain exactly how he would leave the European Union, and why police were called to a noisy row with his girlfriend last week.
Mr Johnson broke cover in a series of broadcast interviews on Monday night and Tuesday, when he refused to discuss his private life but gave further details on his Brexit strategy.
He followed up with a letter to Mr Hunt, challenging him to commit to keeping to the latest delayed Brexit date of 31 October, "deal or no deal".
"We must not kick the can down the road again," he wrote in the message posted on Twitter.
Oct 31, 'do or die'
Mr Johnson was a leading campaigner for Brexit in the 2016 EU referendum, whereas Mr Hunt backed staying in the bloc - a disadvantage among the largely eurosceptic Conservative membership.
But Mr Hunt hit back at Mr Johnson's letter by asking again why his rival refused to attend a Sky News TV debate that had been planned for Tuesday evening.
"Why not turn up to Sky tonight and I'll give you full and frank answers?" he tweeted.
Critics of Mr Johnson question his position on Brexit, asking how he can maintain his coalition of die-hard eurosceptic Conservative MPs and moderates alike.
In an interview with TalkRadio, he said he would keep the 31 October date "do or die, come what may".
Mr Hunt said that was a "fake deadline" likely to trigger a general election.
Mr Hunt would delay Brexit if a deal was within sight but leave on 31 October "as a last resort" if it was not, he told BBC television.
He pitched himself as more likely to wrangle a deal out of Brussels.
"If you choose someone where there's no trust, there's going to be no negotiation, no deal. And quite possibly a general election which could mean we have no Brexit either," he said.
The Conservatives currently command a majority of four in parliament's lower House of Commons.
Around a dozen Conservative MPs are also said to be ready to bring down a Johnson government in order to stop a "no deal" scenario.
Both Mr Hunt and Mr Johnson are hoping to renegotiate the divorce deal May struck with the EU, which parliament has repeatedly rejected - even though Brussels says this is not possible.
Mr Johnson said on Tuesday he wants to keep "the best bits", protecting the rights of EU expats and setting up a post-Brexit transition period while removing its arrangements for the Irish border.
If that fails, he suggested a "standstill" trade deal under World Trade Organization rules, although this needs EU agreement.
He said Britain would threaten "no deal" and withhold its £39 billion ($50 billion) share of EU liabilities until this is done.
However, the EU has repeatedly said it will not sign any deal that does not include the "backstop" plan to keep open the border between the UK's Northern Ireland and EU member the Republic of Ireland.