The 30-year-old centre-back departs BVB after five seasons at the club, during which he made 130 Bundesliga appearances.
An experienced Greece international, Sokratis' arrival will likely be welcomed by Arsenal fans as the Gunners enter life after Arsene Wenger.
Arsenal's defence has been lacking for some time, overly reliant on a declining Laurent Koscielny, and the addition of Sokratis could go a long way to making their back line more robust.
He is strong in the challenge and boasts the physical prowess to control a penalty area.
Per Jack Pitt-Brooke of The Independent and BBC Sport's David Ornstein, head of recruitment Sven Mislintat played a key role in identifying Sokratis as a target given his links to Dortmund and the deal has been expected for some time:
David Ornstein @bbcsport_david
As per @JackPittBrooke Arsenal close to signing Sokratis Papastathopoulos from Dortmund. Latest acquisition led by Sven Mislintat, continues drive to rebuild defence/squad with experience/leadership. #AFC prioritising ready players & will integrate youth (eg Mavropanos) over time
Arsenal have significant ground to make up if they are to break back into the top four next season, let alone challenge for the title.
Key areas need to be addressed, and Arsenal have already added Stephan Lichtsteiner and Bernd Leno to the squad this summer.
Sokratis becomes Arsenal's third new addition and, although he is far from a fix-all solution, he is a defender of proven quality who should be an upgrade on the Gunners' current options at centre-back.