The prime minister believes that privatising the publicly-owned broadcaster is a “great idea”, according to a report in the Sunday Express.
His alleged remarks come in the wake of a series of rows between Downing Street and Channel 4 over its coverage of the election campaign.
Mr Johnson appears to have boycotted the broadcaster since its head of news Dorothy Byrne described him as a “proven liar” in August last year.
He pulled out of a planned interview and refused to take part in Channel 4’s leaders debate on climate change in November.
The prime minister was replaced by a melting ice sculpture in the studio.
An official complaint was later made to media watchdog Ofcom by the Conservative Party, which claimed that the broadcaster was “conspiring with Jeremy Corbyn”.
It was also reported that the party was threatening to review Channel 4’s broadcasting remit.
In the party’s latest attack, a “well-placed Conservative MP” was quoted as saying: “Boris has privately confirmed that he will definitely privatise Channel 4. He thinks it’s a great idea. There is no business for the public to own it anyway.”
It follows recent reports that the prime minister had vowed to scrap the BBC TV licence fee.
“We are having a consultation and we will whack it,” a Downing Street source was reported as saying.
The Department for Digital, Culture, Media and Sport said it would not comment on what it called “speculation”.