Redknapp managed the Spurs star during their time together at White Hart Lane.
He told talkSPORT: “Yeah, I think he will. Gareth won’t want to leave, he’s happy at Tottenham.
“He’s not an over-ambitious lad. He wants to be successful obviously but he’s not someone I can see moving to Real Madrid at this stage in his life.
“His football is going great, he’s PFA Footballer of the Year and loved in this country by football fans who know the game.
“He’s very happy at Tottenham and I can see him staying for at least another year. Eventually he’ll feel it’s time to give it a go abroad, somewhere like Barcelona or Real Madrid, because he is capable of playing anywhere.
“He is the third best player in the world at the moment, behind Lionel Messi and Cristiano Ronaldo.”