Mr Johnson has not answered questions directly in recent days, having not made a public appearance in the opening days of the race for 10 Downing Street.
His campaign team have instead pointed to previous comments by the ex-foreign secretary.
But former Conservative chief whip Mark Harper – who launched his own leadership campaign on Tuesday – stressed “all candidates in this race owe the public clear answers”.
He said: “If I’m a Conservative MP in a marginal constituency, I want the person who’s going to be leading my party to answer these questions now.
“I don’t want them to be answering these questions in three years’ time when it’s going to be my marginal constituency that’s up for grabs.
“So I think everyone has to answer those questions, and then my colleagues will make a judgement about how frank and open we’ve been.
“If you’ve got nothing to hide, you won’t mind answering the questions.”
On his own experience of drugs, Mr Harper added: “I haven’t taken any illegal drugs in my entire life.
“I come from a background, frankly, where I was brought up where people didn’t do that sort of thing.
“I don’t get invited to those sort of parties and I don’t hang out in those sort of circles.”
He also agreed with Home Secretary Sajid Javid who, in the wake of Mr Gove’s admission, took aim at middle-class drug users and the impact of their lifestyle on other parts of society.
“There are consequences for that sort of behaviour and it tends to fall on those, perhaps, from less privileged areas of society,” Mr Harper said.
Mr Johnson gave an account of his experience of cocaine while appearing on TV show Have I Got News For You in 2005.
“I think I was once given cocaine but I sneezed and so it did not go up my nose,” he said.
“In fact, I may have been doing icing sugar.”
He later told GQ magazine in 2007: “I tried it at university and I remember it vividly.
“And it achieved no pharmacological, psychotropic or any other effect on me whatsoever.”
Asked about those previous comments in a 2008 interview, when he was standing for mayor of London, Mr Johnson said: “Well, that was when I was 19.
“It all goes to show that, sometimes, it’s better not to say anything.”
Following that interview, Mr Johnson said days later: “To say that I have taken cocaine is simply untrue.
“As I have said many times, I was once at university offered a white substance, none of which went up my nose and I have no idea whether it was cocaine or not.”