The 23-year-old, who netted 24 goals for Stuttgart in 2008/09 to fire the former champions to a third-place finish, had been linked with summer moves to Arsenal, Liverpool and Manchester City.
However, the front-man - who has 23 caps for his country, scoring six times - has opted to continue plying his trade in his homeland, with Bayern aiming to reclaim their crown from Wolfsburg next season.
“I saw no need and had no desire to move abroad,” he said at a press conference in Shanghai as part of Germany's tour of Asia.
“I am moving because I want to take the next step in my career and Bayern are the German club who can hold their own in Europe.
“I had offers from Italy, England and Bayern and I only needed two days to decide.
“Bayern have been interested in me for the last two years, even after a bad European Championship, and this is also why I didn't go abroad.
“I'm a sportsman who wants to win everything possible and this chance is better in Munich than in Stuttgart.
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