Jeremy Hunt has added his voice to calls for Tory leadership rival Boris Johnson to answer questions about a row with his girlfriend which led to police being called to their address.
Mr Hunt said someone who wants to be PM “should answer questions on everything”.
Cabinet minister Liam Fox said it was better to explain what happened than allow it to become a “distraction”.
But MPs campaigning for Mr Johnson argue that it is a private matter.
Leadership frontrunner Mr Johnson refused to answer questions on the issue on Saturday at a Conservative Party hustings held as part of the contest to replace Theresa May as leader and ultimately prime minister.
It comes after a neighbour called police and recorded a heated row at the home Mr Johnson shares with his partner, Carrie Symonds, in Camberwell, south London.
Defending his actions, neighbour Tom Penn told the Guardian he had been worried about his neighbours’ safety, adding: “I hope that anybody would have done the same thing.”
He said he began recording from inside his flat, after he heard “slamming and banging” in the early hours of Friday.
In the recording – heard by the Guardian, but not by the BBC – Ms Symonds reportedly could be heard telling the Tory MP to “get off me” and “get out of my flat”. Asked about the issue, Mr Hunt told Sky News: “I think someone who wants to be PM should answer questions on everything, but I’m not going to comment on character.”
But the foreign secretary also said he thought the story about Mr Johnson’s row with his girlfriend was “irrelevant to the leadership debate” because the country was in “such a serious situation” over Brexit.
“What happens in people’s personal lives is really a matter for them.
“What people care about is who is going to be the wise prime minister who navigates this country out of the biggest constitutional crisis in our lifetimes.”
Mr Hunt went on to criticise Mr Johnson for “shying away” from doing interviews and not answering “difficult” questions about Brexit.
“What we need is for him to engage properly,” he said.
Mr Hunt also said Mr Johnson’s premiership would not last if he was not straightforward with MPs about what his Brexit plans were.
“People are beginning to wonder about the coalition of people supporting him… what has he being telling them… which is the Boris we will get as prime minister?”