The Belgium international has been granted permission to travel to Istanbul, where he will undergo a medical before completing his move to the Turkish Super Lig, in a deal reported by the Evening Standard to be worth around £20m.
Arsenal confirmed the agreement in a club statement.
"We can confirm that we have reached an agreement with Besiktas for the permanent transfer of Leandro Trossard.
"With a transfer fee agreed, Leo, 31, has been granted permission to travel to Istanbul to undergo a medical and complete the formalities of his move to the Turkish Super Lig club.
"We will provide a further update once the transfer has been completed."
Trossard joined Arsenal from Brighton in January 2023 and became a fixture in Mikel Arteta's squad, making 174 appearances and sharing left-wing duties with Gabriel Martinelli. He played his part in last season's title triumph, which ended the club's 22-year wait for a Premier League crown, and had entered the final 12 months of his contract at the Emirates Stadium.
The winger took his time over the decision, having been on World Cup duty with Belgium in the United States when Besiktas opened talks with his camp, and is expected to sign a three-year contract in Istanbul with the option of a further year.
His departure is now set to accelerate Arsenal's summer transfer business, with the champions keen to refresh their wide options. Aston Villa's Morgan Rogers and Club Brugge winger Christos Tzolis are among the names linked with the vacancy, while Besiktas, managed by Sergen Yalcin, are also reported to be in talks over a move for Trossard's compatriot Romelu Lukaku.
Arsenal say a further update will follow once the formalities are finalised, at which point Trossard will bring the curtain down on a spell that yielded a Premier League title and a string of decisive goals in north London.