The 29-year-old England international has been out of favour under Spurs boss Ange Postecoglou, making only four Premier League appearances this season.
Spurs have also completed the signing of Genoa centre-back Radu Dragusin for around £25m.
"This move is a dream come true for me, because as a child you want to play for a club like Bayern one day," said Dier.
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More to follow.