The 29-year-old German international had three years left on his deal but will now stay a further season until 2023 at the club which is presided over by Florentino Perez, reports Efe news.
The former Bayern Munich player joined Madrid in 2014, made 233 appearances and won 12 trophies during his five-year spell, including three UEFA Champions League titles, three UEFA Super Cups and a La Liga crown.
This is the first move that Madrid has made so far this summer as head coach Zinedine Zidane undertakes to rebuild his squad after a disappointing season in which it was eliminated from the Champions League and the Copa del Rey, while it only managed to finish third in La Liga's end-of-season standings.
Madrid ended its La Liga season on Sunday with a 2-0 loss to Real Betis to be 19 points away from leader Barcelona.
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