The UK's Boris Johnson challenged his rival for Britain's leadership on Tuesday to deliver Brexit by the end of October whatever happens, only to be accused once again of trying to avoid scrutiny.

Johnson, the former foreign secretary and ex-mayor of London, is the favorite to succeed Theresa May as leader of the governing Conservatives and therefore as prime minister.

But his rival Jeremy Hunt, the current foreign secretary, has accused him of being a "coward" for refusing to face him in a one-to-one TV debate.

The pair are battling for the votes of an estimated 160,000 Conservative party members, with the winner declared on July 23, and taking office on July 24.

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 EU, and why police were called to a noisy row with his girlfriend last week.

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 Hunt, challenging him to commit to keeping to the latest delayed Brexit date of October 31, "deal or no deal."

"We must not kick the can down the road again," he wrote in the message posted on Twitter.

Johnson was a leading campaigner for Brexit in the 2016 EU referendum, whereas Hunt backed staying in the bloc - a disadvantage among the largely eurosceptic Conservative membership.

But Hunt hit back at Johnson's letter by asking again why his rival refused to attend a Sky News TV debate that had been planned for Tuesday evening.