Kimmich, 19, is currently on loan at 2.Bundesliga side Red Bull Leipzig from the Mercedes-Benz Arena and will move to Bayern at the end of the current season.
Having come through the youth system at Stuttgart, Kimmich is yet to feature for the Bundesliga side but has been a regular feature for Leipzig in their rise up the German pyramid.
Kimmich - who has represented Germany at a number of youth levels - made 26 appearances during last season's 3.Bundesliga campaign as Leipzig sealed promotion.
He has made 13 league appearances this term - helping Leipzig to seventh in the second tier - with Kimmich a former Under-19 European champion with Germany.
"Kimmich is a desired player of [sporting director] Matthias Sammer and [coach] Pep Guardiola," read a section of Bayern's statement.
The transfer fee between Bayern and Stuttgart has been left undisclosed.
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