Kjaer suffered ligament damage in victory against Genoa in midweek.
Sportmedia reports the Denmark international damaged the ligaments in his left knee and, therefore, will undergo surgery in Milan on Friday.
Kjaer was stretched off the pitch after four minutes and underwent medical examinations in Milan on Thursday morning.
Stefano Pioli admitted being worried by Kjaer's condition and the player arrived at the Clinica Columbus in Milan on crutches.
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