The Germany international was injured during Real's 2-0 home defeat by Real Sociedad on Saturday which left his side fifth in La Liga, 10 points behind leaders Barcelona.
"(Kroos) has been diagnosed with a grade two tear of the adductor muscle in his left leg. His recovery will be monitored," said Madrid in a statement.
Kroos is set to miss the Copa del Rey last 16 games against Leganes plus Liga matches with Real Betis, Sevilla and Espanyol.
Madrid, who were whistled off by their fans at a half-empty Santiago Bernabeu on Sunday, are already without the injured Gareth Bale, Marcos Llorente, Marco Asensio and Mariano Diaz.
(Reporting by Rik Sharma; editing by Ken Ferris)
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