Clarkson has been criticised for jokes about Mexicans and other national groups during previous editions of Top Gear.
But Mr Cohen revealed that Clarkson had rejected criticisms of his language when the pair had discussed the director of television's unhappiness with the presenter's behaviour.
"He doesn't see a problem with some of the language he has used," said Mr Cohen. "He feels differently about it from me."
Appearing later on Richard Bacon's show on BBC Radio 5 Live, Mr Cohen said he had had several "frank" conversations with Clarkson but refused to say whether the star was on a final warning.
He said: "I've been very clear with what I think is acceptable on the BBC."
Asked about the N-word incident, Mr Cohen said: "Regardless of it being on television or not, I don't want that stuff going on when we are on production, when you're representing the BBC."
In a letter to the Guardian this month, Mr Cohen said he disagreed with Clarkson's language but insisted the Top Gear host was not a racist.
Clarkson apologised in May over the N-word incident, which occurred in an unaired episode of the BBC motoring show that was published on the Mirror website.
- Independent