The 26-year-old had entered the final year of his deal, which included an option for a further season, and talks over a £3m move to West Ham earlier in the summer failed to progress, with the Championship side moving slowly and Nelson reluctant to drop out of the top flight.
Rather than let the situation drift, Arsenal opted to release him without a fee, a move The Athletic reports was driven primarily by the need to clear space on the wage bill.
Nelson had fallen outside Mikel Arteta's plans after spending last season on loan at Brentford, where persistent injuries limited him to just 13 appearances.
Nelson leaves having made 90 first-team appearances for the club he joined as an academy player, scoring eight goals and adding nine assists, and picking up an FA Cup winner's medal in 2020 along with two Community Shields.
His time in north London included loan spells at Hoffenheim, Feyenoord, Fulham and Brentford, with his senior career never quite matching the promise he showed coming through the ranks. He is now free to negotiate with clubs in England or abroad and is expected to take time weighing up his next move.
His exit is the first of what Arsenal hope will be four departures before Monday's deadline. Gabriel Martinelli is closing in on a club-record fee in the region of £50m to join Al-Hilal, while Gabriel Jesus and Fábio Vieira both remain on the transfer list, having similarly been frozen out of Arteta's first-team plans.
The clearout comes as Arsenal continue to push for a deal to sign Atlético Madrid striker Julián Álvarez, with the departures freeing up both wage room and squad space to accommodate a potential arrival before the window shuts.