De Gea won the Premier League's Golden Glove award this season for the most clean sheets.
"They're [United players] all patting De Gea on the back there…I'd move him on quickly. He wouldn't be for me, he's not going to get Manchester United back winning trophies, absolutely not.
"The idea that the players are patting him on the back is ridiculous, that's his job."
Keane continued: "No, you wouldn't keep him as a No. 2. He wouldn't want that, he wouldn't stay for that.
"He's done okay, he's won a few Player of the Year awards – but what big deal is that? It's about what the team is achieving.
"I think he's made far too many mistakes over the years. He was trying to leave the club for a few years ago and go to Real Madrid."