The 31-year-old made 174 appearances in all competitions for Arsenal, scoring 36 goals after joining from Brighton & Hove Albion in January 2023.
Besiktas will reportedly pay around £17 million for Trossard, whose contract was due to run until 2027 with an option for a further year.
Arsenal said Trossard had been given permission to travel to Istanbul for a medical.
Trossard scored six goals and provided six assists in 31 league appearances as Arsenal, under Mikel Arteta, ended their 22-year wait for a league title.