Media reports in Germany put the transfer at 35 million euros ($40 million) for the 22-year-old who can play at right-back or centre-back.
"He is young and a world champion, so we are happy and proud to have won over such a player," said Bayern's sports director Hasan Salihamidzic at their winter training camp in Doha.
Pavard's status skyrocketed last summer when the centre-back scored the goal of the World Cup with a stunning net-busting strike in a 4-3 last-16 win over Argentina that set France on their way to the Russia 2018 title.
Pavard has been at struggling VfB Stuttgart, who are currently third from bottom in Germany's top flight, since 2016 after he joined from French side Lille for five million euros.
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