The 29-year-old will become new Arsenal boss Unai Emery's second signing of the summer window, having already brought in the experienced Stephan Lichtsteiner.
According to Bild, the London club is set to pay a transfer fee of around €19m to secure the highly-rated defender, whilst the player is due to double his wages.
It is reported that the Greece international will be set to earn £84,000 per week at the Emirates. The German publication also state that Sokratis' move to the Premier League is almost complete, with a medical the only element of the transfer still to be carried out.
The player's father had already told Greek publication Sport 24 that his son was set to become an Arsenal player. Last season, the defensive star played 30 games for Dortmund in the Bundesliga, and scored on two occasions.
Much like Emery's first signing in Lichtsteiner, Sokratis offers Arsenal a solid defensive option with plenty of footballing experience to help bolster a back line that has been criticised on several occasions in recent years.
The central defender has an impressive 79 senior caps for the Greece national side, and has scored three senior international goals.
One of those goals memorably came in the 2014 World Cup finals, when he netted a late equaliser in second half injury time against Costa Rica to keep Greece in the match. They ultimately lost 5-3 on penalties.
At the Emirates, Sokratis will also link up with two former teammates at Borussia Dortmund in Pierre-Emerick Aubameyang and Henrikh Mkhitaryan - both of whom joined the club in the January transfer window.