Carragher, 35, announced he will retire at the end of this season.
Former Liverpool and Wales ace Rush says the defender should stay at Anfield in a coaching role before taking the reigns himself.
He said: “Hopefully one day he will become a manager and hopefully that will be of Liverpool.
“I’m pretty certain he’s got the qualities to be that.
“He’s a winner and if he can instill that in the players that’s when you become a good coach.”