The 22-year-old, who joined Bayern last summer after rejecting a move to Old Trafford, suffered ligament damage during the first half of Saturday's 3-3 draw with Hoffenheim.
A Bayern statement said: "The first diagnosis for Thiago is a suspected partial tear of the ligament. He will most likely miss both quarter-final legs against Manchester United."
Coach Pep Guardiola told reporters: "Thiago Alcantara will be out for about a month most likely. That is really bad news. We will miss him."
Alcantara has made 20 first-team appearances for Bayern, scoring two goals in the process.