Along with Mock The Week, the BBC panel show has faced complaints over its jokes about the Conservatives and Brexit.
But Ian, who has been a captain on HIGNFY with comedian Paul Merton for nearly 30 years, said critics of the show believe they are “Momentum-inspired, lefty lunatics”.
Speaking at the Cheltenham Literature Festival to mark his 60th birthday, Ian said: “I feel there was a moment where again in this sort of polarised world, somehow the director general, or someone who was briefing others, got it into their heads that any criticism of the government on a panel show suggests this is a bunch of sort of Momentum-inspired, lefty lunatics.
“I just now quote from the Daily Telegraph, who have really got it in for Boris, and ask just how right-wing you would like us to be.”
Ian, who is the editor of satirical magazine Private Eye, continued: “We’re sort of around the Spectator and the Telegraph editorial at the moment. There is quite a lot of that sort of nonsense about.
“We could be fired. That would be good.”
Ian’s comments come after it was reported new BBC director general Tim Davie was considering axing a number of comedy shows to tackle perceived bias.
Earlier this month, Mock The Week host Dara O’Briain dismissed the criticism, calling the debate about whether his show was too left-wing “frankly mind-numbing”.
He told the PA Media news agency: “When a story pops up on Mock The Week, no-one is going, ‘How can I bend this to an agenda?’
“You’re going, ‘What’s the gag? What is the joke here? Where can I find the joke?’”
During his speech, Ian said he “occasionally worries” about being sacked from Have I Got News For You, recalling one particular moment that he and Paul Merton went against the wishes of former BBC director general John Birt.
He said: “There was a point where Peter Mandelson was involved in both a sort of mortgage scandal and he was outed by Matthew Paris as gay.
“The director-general, who was a friend of Mandelson, sent round a memo to all BBC programmes saying there will be no mention of Peter Mandelson on this programme at all and nothing that happened in Peter Mandelson’s life will be mentioned.
“I was asked a question and I said, ‘Interestingly it turned out this week that Peter Mandelson turns out to be a home… owner’ at which point Paul Merton said, ‘Yeah, but why shouldn’t gay people have mortgages’.
“Again, I think we’ve got form on this front.”
Have I Got News For You continues on Friday at 9pm on BBC One.