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"Medical tests confirmed that our player suffered a grade two hamstring tear in his left leg," Atletico said in a statement on Monday without putting a timescale on his return.
Savic is likely to miss league matches against Huesca, Getafe and Real Betis, as well as the Copa del Rey last-16 second leg clash with Girona.
Atletico beat Levante 1-0 and remain five points behind La Liga leaders Barcelona.