"It's like football clubs. No-one is bigger than the club," said Cohen. "No one show or person is bigger than the BBC and that includes me. I found (the language) entirely unacceptable."
Clarkson was criticised by the BBC Trust after a Top Gear special in which he referred to "a slope" on a bridge, which was being crossed by a Thai man.
He previously expressed regret for a an outtake from Top Gear in which he could be heard to utter the N-word while reciting the rhyme "Eenie Meenie Minee Mo".
Clarkson has been criticised for jokes about Mexicans and other national groups during previous editions of Top Gear.
But 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 Cohen. "He feels differently about it from me."
Cohen said he did not think it was appropriate for the hit show to be transferred from BBC2 to BBC1, in line with a similar move made by The Great British Bake Off
Discussing the future of BBC2, Cohen said the channel would be broadcasting a series on the history of Black Britain and showing what he described as "Britain's first transgender comedy", called Boy Meets Girl.
In the Autumn, BBC2 will broadcast Castles in the Sky, a drama on the invention of radar, starring Eddie Izzard.
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