The Gunners were understood to have made contact with the Greek defender's representatives earlier this week, and talks have progressed over a deal which would keep him at the club until the end of the 2019-20 season.
German outlet Bild claims that Arsenal are now close to finalising the two-year contract, with Sokratis set to become the club's first signing of the post-Arsene Wenger era.
The 29-year-old is well known by Arsenal's head of recruitment Sven Mislintat, who helped to take him to Dortmund from Werder Bremen in 2013.
The Gunners are thought to have made a new defender one of their main transfer priorities this summer, having conceded 51 Premier League goals this season - their highest tally in the league since 1983-84.
Sokratis made 30 Bundesliga appearances for Dortmund this season.