The 32-year-old, who has not started a game since March, failed to make Arsenal’s Premier League and Europa League squads this season and has been frozen out by manager Mikel Arteta.
His current deal was set to expire in the summer but the centre back has agreed to end his time with the club earlier than planned.
Arsenal’s technical director Edu said: ‘On behalf of Mikel, our coaches, players and everyone at the club, I would like to thank Papa for his contribution to the club.
‘He has been an important part of our group and has been a model professional throughout. The negotiations to end his contract have been collaborative and we wish Papa and his family well for the future.’
The defender joined Arsenal in a £17million deal from Borussia Dortmund in 2018 and has made 69 appearances for the north London side.
His last appearance was as a last-minute substitute in the FA Cup final in August, which Arsenal won 2-1.
The termination of his contract would leave Sokratis free to agree a deal with another club.
The Greek international has had interest from Genoa, who he played for from 2008 to 2010.
Napoli and Fenerbache have also been linked with the centre back. A move to Fenerbache would see Sokratis link-up with former team-mate Mesut Ozil.