The 22-year-old, who joined Bayern last summer from Barcelona, suffered suspected knee ligament damage during the first half of Saturday’s 3-3 draw with Hoffenheim.
A Bayern statement read: “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, whose side retained their Bundesliga crown in midweek, told reporters: “Thiago Alcantara will be out for about a month most likely.
“That is really bad news. We will miss him.”
Meanwhile, United boss David Moyes insisted that his players “incredibly focused” heading into the first leg at Old Trafford on Tuesday.
“It’s been a difficult season for us but we can go a long way to doing an awful lot better if we can get a result on Tuesday,” said Moyes, whose side beat Aston Villa 4-1 on Saturday.
“The players are incredibly focused and I am really looking forward to the game.”
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