Fernandes, 31, has had a remarkable season for Man United, breaking the Premier League assist record with his 21st in their 3-0 win over Brighton on the final day.
Keane, however, hasn’t been impressed, and accused him of prioritising the individual assist records over team success.
The current Man United captain then hit back at Keane on the Diary of a CEO podcast, saying he is lying and that Fernandes never made the comments he’s been accused of making.
Fernandes told host Steven Bartlett: "I've always said I don't mind criticism. I always take criticism from everyone and I never reply to anything.
“People have an opinion, they think it's good, bad or whatever. What I don't like is when people lie about things.”
He added: "This case that you said about Roy Keane, basically what he said is a lie because either he saw some other interview or he can't say that I said one thing that I've just not said.
“I accept his criticism, whether he likes me as a player or not, likes me as a person or not. But what I don't like, is that he puts words in my mouth that has not been said."
Keane has now taken to Instagram to seemingly respond, posting an AI image with the caption: "Too much attention makes a donkey think he's a lion."