The Labour leader's remarks came as a series of shadow cabinet ministers attempted to turn fire on the media after internal divisions overshadowed key policy announcements at the party's annual conference.
In a video published shortly after the programme was broadcast on Sunday, Mr Corbyn highlighted a report The Sunday Times.
The newspaper carried allegations that Boris Johnson failed to declare potential conflicts of interest over a close friendship with an American model turned technology entrepreneur.
It was claimed Jennifer Arcuri, who moved to London seven years ago, was given £126,000 of public money and was afforded privileged access to three foreign trade missions Mr Johnson led.
"There was something important in one of the Sunday papers today about an alleged abuse of power and misuse of public funds by Boris Johnson before he became prime minister," Mr Corbyn said in the video posted on Twitter.
He continued: "But do you know what? Lots of the media have barely touched it. Incredible, isn't it? This is about the man who is the prime minister of our country.
"I was on the BBC's flagship news programme this morning and I was asked about a range of issues. That's fine, it's right, it's justified, it's how our democracy works.
— Jeremy Corbyn (@jeremycorbyn)September 22, 2019
But referring to the appearance of shadow foreign secretary, Dominic Raab, on the BBC's Andrew Marr programme, he added: "But I was followed by Boris Johnson's deputy who was asked nothing about these allegations.
"This is how the establishment works. They close ranks. They put privilege first. Things have to change."
Responding to Mr Corbyn's comments, Mr Marr said on Twitter: "I was genuinely pleased that, on what cannot have been an easy morning for him, JC turned up for a live interview. And of course I always welcome advice on how to do my job."
A BBC spokesperson added: "The story was covered during the programme on the news review. Every interview requires editorial decisions and Andrew focused on the interview with Dominic Raab on Thomas Cook, Iran, the Supreme Court decision and Brexit."
Ms Arcuri, now 34, was quoted by The Sunday Times as saying: "Any grants received by my companies and any trade mission I joined were purely in respect of my role as a legitimate businesswoman."
She did not comment on the nature of her relationship with Mr Johnson, and Downing Street declined to comment.
Elsewhere, Diane Abbott, the shadow home secretary and close ally of the Labour leader, also lashed out at the media at a fringe event hosted by The Guardian's editor Katherine Viner.
She said: "This must be my 30th Labour party conference and I cannot remember one when the media weren't running around like headless chickens talking about Labour splits. Every year. If the NEC had a discussion about whether to have chocolate biscuits or ginger nuts, people like you would say, 'Oooh Labour splits".