The Greece international rejected the overtures of Manchester United according to an interview given by his father to Greek radio and will be announced as Arsenal player when the transfer window officially opens next month.
Sokratis is expected to sign a three-year deal with the north London club.
After spending five years at the Westfalenstadion, the Greek defender was keen for another challenge and was about to enter the last year of his contract.
With no agreement between player and club in sight, Dortmund decided to cash in on the centre-back rather than lose him for nothing next summer.
Arsenal tried to sign Sokratis last summer but talks with Dortmund broke down. However, landing Pierre-Emerick Aubameyang, 28, and Henrikh Mkhitaryan, 27, made the Emirates a more attractive destination for former BVB players.
The signing has the hallmarks of new Arsenal head of player recruitment Sven Mislintat who joined from Dortmund late last year. The connection between the German and his former club is seemingly unshakable.
Arsenal are desperate to strengthen their back four with the Achilles injury suffered by Laurent Koscielny forcing them to act quickly.