Modric's current deal runs out in June 2021 and he stated in his autobiography, released earlier this summer, that he feels he has at least two more years left in him at the club he has called home since 2012.
Sport says Modric will only be contracted to play on for Los Blancos until the summer of 2022 due to his age.
Modric joined Real from Tottenham in 2012 and has made 347 appearances for the club, scoring 22 goals.
"I'm certain that I can play at a high level for two more years, then we shall see," he said in his book. "I would like to end my career at Real Madrid, but it will also depend on the club. Of course, I will then do the course to become a coach."
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