"Whether it's complacency or cowardice, it's not a good look for Boris to be ducking debates with me. Because if you want the job, you need to show up for the interview... So don't be a coward Boris - man up and show the country you can cope with the intense scrutiny the most difficult job in the country will involve", Jeremy Hunt said.
The statement comes after Johnson decided not to appear on Channel 4's Tories leadership debates earlier in June. He, however, challenged Hunt to a debate with him on Tuesday.
The ex-foreign secreatry has previously faced public scrutiny since police were called by his neighbour over an alleged explosive row between Johnson and his girlfriend Carrie Symonds.
The race in Britain's Conservative Party began after Prime Minister Theresa May announced her resignation last month amid a deadlock over Brexit. She will step down as caretaker PM on 22 July to be replaced by one of two candidates chosen by the party's members after a postal ballot.