The 18-year-old made his Bundesliga debut in February for Cologne, while still aged 17, and was part of the Germany team which finished third at last year's under-17 World Cup in Mexico.
"We are very happy to have Mitchell Weiser, one of Germany's most talented future prospects, in our squad," said Bayern's director of sport Christian Nerlinger.
The son of former Wolfsburg and Cologne midfielder Patrick Weiser, the teenager said he was looking forward to competing for a place in Bayern's first-team next season.
"As a junior, when you get the chance to play for Bayern Munich, you have to take the challenge," he said. "This is a great chance for me."
Weiser is Bayern's fifth new signing for next season along with Moenchengladbach defender Dante, striker Claudio Pizarro from Bremen, midfielder Xherdan Shaqiri from Basel, goalkeeper Tom Starke from Hoffenheim.
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