The 31-year-old leaves Arsenal after two-and-a-half successful seasons, having joined the London club from Brighton in January 2023.
During his time at the Emirates Stadium, Trossard made 174 appearances in all competitions and scored 36 goals. He became an important member of Mikel Arteta's squad, regularly delivering in big moments, including the winning goal in a 1-0 victory over West Ham last season.
The Turkish club are reported to have agreed a deal worth 18 million euros, with a further two million euros possible through performance-related bonuses.
Besiktas also have the option to extend Trossard's contract by an additional year.
The Belgium international said his respect for Arsenal played a major role in his decision to continue his career outside England.
"One of the reasons I chose Besiktas over some clubs in England is because I didn't want to be playing against Arsenal," Trossard said.
"This club means too much to me, and I didn't want to find myself on the opposite side."
New Crystal Palace manager Pierre Sage believes his team can reach "another level" after beginning their pre-season campaign with a convincing 5-1 victory over Swindon Town.
Michael Carrick believes Andrey Santos showed plenty of promise on his Manchester United debut despite the club's 1-0 pre-season defeat to Wrexham in Helsinki on Saturday.
Elijah Upson, the son of former Arsenal and England centre-back Matthew Upson, is set to join the Gunners after leaving North London rivals Tottenham, with an official announcement expected in the coming days.
Manchester United remain determined to challenge for the biggest prizes, with director of football Jason Wilcox insisting the club's ambitions have not changed despite an ongoing rebuilding process.
