According to Bild, the Greek centre-back has been targeted by Arsenal's head of recruitment Sven Mislintat, who has made recruiting defensive reinforcements his top priority for the summer.
The Gunners are still searching for their new manager, with former club captain Mikel Arteta reportedly close to being appointed. Mislintat, who knows Sokratis from his time working at Dortmund, is determined to bring in players who will give Wenger's successor the means to succeed in their first season in charge.
The report by Bild suggests that Sokratis, who has been at Dortmund since 2013, will sign a two-year deal with Arsenal. The German side will look to replace the defender by moving for French Under-21 international Abdou Diallo, who has attracted a number of suitors following an impressive season for Mainz.
Sokratis' contract at Dortmund expires next summer and with the two parties struggling to agree on a new deal, Arsenal could get their man for a relatively inexpensive £17m. This would represent good value for the 29-year-old defender, who has been capped 17 times by Greece.
The signing of Sokratis would represent an attempt to shore up Arsenal's notoriously leaky back line, the side's defensive frailties having been the subject of criticism for the past few seasons.
In addition to Arsenal's pre-existing defensive woes, club captain Laurent Koscielny will miss the first few months of the new season through injury, making the signing of a new centre-half all the more essential for the Gunners.