The Germany international is said to have agreed a four-year contract with the club and will finalise his move following the successful completion of a medical on Tuesday.
Dortmund will reportedly receive a fee that could rise to €20m in add-ons.
Ginter, a DFL-Supercup and DFB-Pokal winner with Dortmund, will be presented at a news conference later on Tuesday.
He made a total of 42 appearances in all competitions under Thomas Tuchel last season, including the full 90 minutes in their Pokal final triumph over Eintracht Frankfurt.
BVB sporting director Michael Zorc confirmed via the club's official website: "Matthias Ginter approached us with his wish to leave the club because of the competition in our defence, and we met his wish."