In a statement, Milan said that Germany defender Thiaw underwent tests on Wednesday morning which showed a "severe musculotendinous tear of the biceps femoris in his left thigh".
Milan did not say for how long Thiaw would be out of action but added that the 22-year-old would have further tests on the injury "in 7-10 days".
Italian media report that Thiaw will likely be unavailable for around two months.
Thiaw pulled up early in the second half of Tuesday's 3-1 loss to Dortmund which left Milan on the brink of early elimination from Europe's top club competition.
His injury is the 25th suffered by Milan this season, after the seven-time European champions had already headed into the clash at the San Siro against Dortmund with eight players on the treatment table, including star winger Rafael Leao.
Milan are bottom of Group F and in order to reach the last 16 must win at Newcastle United in their final match next month and hope leaders Dortmund beat Paris Saint-Germain.
Stefano Pioli's team are third in Serie A, six points behind leaders Inter Milan, and host Frosinone on Saturday.
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