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.
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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)
