The 26-year-old striker has scored 112 times in 235 games for Los Blancos since making a €35 million (£27.9m) move from Olympique Lyonnais in July 2009 and also has 66 France caps to his name.
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Zidane is confident that his compatriot can play an increasingly key role for club and country and start to be seen in the same light as Madrid's traditional leaders.
"I would like Karim to take the initiative at Madrid," Carlo Ancelotti's assistant told reporters.
"He's the kind of player who doesn't talk much but he has his feet on the ground. He must move up a level to become a leader.
"As in the France team, he can inherit the captain's armband with his club.
"I hope he meets the challenge next season. Karim can talk to the group, rather than have others do it for him."