The 35-year-old sustained the problem in the second half of last weekend's draw with Bologna and was forced to leave the pitch due to the knock.
Yepes underwent further examination on Wednesday and has been diagnosed with damaged ankle ligaments.
Milan have announced on their official website that the Colombian centre-back will have to undergo surgery to rectify the injury and will consequently be out of action for up to 10 weeks.
It's not yet known whether Yepes will be operated on in Italy or in Amsterdam.
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