The full-back was caught by a crude challenge from Rodrigo De Paul in Sunday's 2-1 loss to Udinese and was forced off the pitch soon afterwards.
Montella slammed the decision to award the Udinese forward only a yellow card for the foul, with Milan's frustrations compounded when De Paul went on to score the winning goal.
Reports in Italy have suggested that De Sciglio could now be sidelined for a number of weeks, and although Milan refused to put a timescale on his recovery, they confirmed that further tests will be required to determine the extent of what looks to be something of a serious problem.
A Milan statement added that midfielder Giacomo Bonaventura will undergo further tests this week to determine the extent of an abductor injury.
Left-back De Sciglio has recaptured some of his best form under Montella this season, making 19 appearances in all competitions.