The Whitecaps said the transfer fee and add-ons for the 17-year-old Canadian could total more than $22 million, the most received by an MLS club.
Davies, who was born in Ghana, has scored three goals in 20 appearances for the Whitecaps this season.
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"Alphonso Davies is a very big talent," Bayern sporting director Hasan Salihamidzic told the club's website.
"At the age of 17, he offers a lot of promise for the future. Alphonso already possesses great ability, which is why a lot of top clubs were keen on him. I'm delighted he's chosen FC Bayern... "
Davies, who turns 18 in November, has signed a contract with Bayern until June 2023 and will officially join them in January following the conclusion of the MLS season.
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