The 28-year-old is out of contract at Paris Saint-Germain at the end of June, and will leave the Ligue 1 side after four seasons.
According to BILD, Meunier is almost certain to be joining BVB and an announcement could come before the season ends.
Several clubs made offers to Meunier, but his compatriots Axel Witsel and Thorgan Hazard helped convince him to pick Dortmund.
Dortmund haven’t given up on extending Hakimi’s loan, but even if he stays it won’t change plans to sign Meunier.
Meunier has made 27 appearances for PSG in all competitions this season, scoring one and assisting two goals.