Last week Johnson apologized for any offence his words had caused, but also claimed they had often been taken out of context.
"Where is Boris? If his team won't allow him out with five fairly friendly colleagues, how is he going to deal with 27 European countries?" he said.
Rival Rory Stewart said Johnson's plans for Brexit would come "off the rails" once subject to detailed examination.
"Boris has run a disciplined campaign and is nearly certainly going to be our next prime minister", Mr Hancock wrote in the Sunday Times. I don't even know what he believes. He won't talk to you.
Johnson has committed to the upcoming BBC debate, only his second public appearance in connection to his leadership campaign.
Mr Leadsom was a leading Brexiteer voice int he Government and her endorsement of Mr Johnson raises questions about the future of fellow hardliner Dominic Raab, who is struggling to remain in the contest.
In addition to Johnson, Javid and Stewart, other contenders include Foreign Secretary Jeremy Hunt, Environment Secretary Michael Gove and ex-Brexit Secretary Dominic Raab.
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We can stop a no-deal Brexit much more easily than that.
Raab said the centre-right Conservatives would be unable to win a general election if they do not take Britain out of the European Union by the deadline.
"And, of course, we all remember that at the end of Hamlet, he isn't the king".
Three years on from the referendum on the UK's European Union membership, Raab said lawmakers opposed to Britain leaving the bloc were trying to block Brexit from ever taking place. "What's really scandalous about this is the way people have been trying to sabotage the will of the people and break their promises", he told Sky News.
"We gave people a decision. Parliament is our democratic representatives and they will meet regardless of what the prime minister wants", he said to applause from the audience. "The big mistake we made in these negotiations is taking no deal off the table and when we start ruling things out we only weaken our chances of getting a deal".
"It's not impossible to do it before October 31 but it will be hard", Hunt told BBC television.
International Development Secretary Rory Stewart said shutting down parliament was "undemocratic" and "disturbing" and would not work.
The victor of the members' vote will be announced in the week beginning July 22 and then take over from Theresa May as prime minister.