The hugely talented 22-year-old Spain international, who joined in the close season from Barcelona, damaged ligaments in his right ankle in the second half of their 2-0 win over Nuremberg on Saturday.
"We will miss him as he will be out for a long time," coach Pep Guardiola said. "He is a special player and very important for the team."
Initially ruled out for seven weeks, Bayern said the player, who was brought in to add versatility and pace to their midfield, could be back in early November.
Thiago will leave the clinic on Tuesday and is expected to use crutches for five to six weeks.
"After that he will then start his rehabilitation training. Guardiola hopes to have 22-year-old back for match action in early November," Bayern said in a statement.
Thiago was one of the treble winners' biggest signings in the close season, joining for a fee of 25 million euros from the Spanish champions on a four-year deal.