The pair were together at Manchester United and Cooke said: "Phil will make a great manager for sure.
"I've always said both him and his brother [Gary] have coaching potential. You could tell when they were young because they were always talking about the game, always watching it, just never stopped thinking about it - he's super professional.
"He'll be a good man-manager and his knowledge and experience will help him be a success."
Cooke added: "I'm not at all surprised he's gone on to be Everton captain.
"He's quiet on the pitch - he's not one of these people who shouts unnecessarily - but you notice him more as a player than as a person. He just gets on with his job and leads by example. That's all you need from a captain â€“ someone who'll knuckle down and work hard. That's what Phil does and why he's been successful."