The England international has faced a negative reaction from Red Devils supporters after former manager Sir Alex Ferguson revealed that he had handed in a transfer request at the end of last season.
Redknapp wrote in his column in The Sun: "The one thing that always strikes you about Rooney is, whatever is happening off or on the pitch, when he pulls on the red shirt he gives everything.
"It's like playing on a Saturday afternoon really is the only thing that matters to him; working, running, scoring. He enjoys the game.
"If he stays at Old Trafford next season he will do exactly the same. The latest dramas that have accompanied him will be forgotten and he will move on. Because of that the supporters will soon be back on side."
Rooney scored 16 goals for United last season.