The health secretary referred to a remark attributed to Mr Johnson last year where he said "f*** business" following concerns from some in industry that a hard Brexit would damage the economy.
In an interview with the Financial Times, Mr Hancock said: "To the people who say 'f*** business', I say 'f***, f*** business'."
Mr Johnson, a former foreign secretary, refused to deny making the comment.
Meanwhile, the race to become the new Conservative leader has a new contender with Braintree MP James Cleverly entering the fray.
In a letter to his constituents in the Braintree and Witham Times, he wrote: "Both the country, and my party, are beset with division.
"We cannot bring the country back together unless the party of government is united, and the party cannot unite if it is led from its fringes.
"I believe the case for Brexit is still valid and I have not wavered in that belief."
Mr Cleverly said a no-deal Brexit was "not his preferred choice", while Mr Hancock told the FT he did not believe no deal would be an option for the next Prime Minister.
He said he believed Commons Speaker John Bercow would block any attempts to impose no-deal without the backing of MPs.
Mr Hancock said: "I think the Speaker would facilitate a majority in the House of Commons who are opposed to no deal in exactly the same way as he did in the run-up to the 29 March."
He added: "The brutal reality is, no deal is not a policy choice available to the next Prime Minister."
Elsewhere, Mr Bercow knocked back suggestions he would step down from the speakership this summer.
Speaking to the Guardian, Mr Bercow said it was not "sensible to vacate the chair" while Brexit was unresolved.
Mr Bercow said: "I've never said anything about going in July of this year. Secondly, I do feel that now is a time in which momentous events are taking place and there are great issues to be resolved and in those circumstances, it doesn't seem to me sensible to vacate the chair."
He added: "If I had any intention to announce on that matter ... I would do so to Parliament first."