Uruguay international Torreira spent the last two seasons on loan at Atletico Madrid and Fiorentina respectively.
The 26-year-old made 89 appearances for the Gunners after moving from Sampdoria in July 2018 at the start of Unai Emery’s reign.
He was an unused substitute when the north London club defeated Chelsea in the 2020 FA Cup final, while he won LaLiga during his temporary switch to Atletico.
“Everyone at Arsenal thanks Lucas for his contribution during his time with us,” read a statement on Arsenal’s website.
“We wish Lucas and his family the best of health and happiness in their new chapter with Galatasaray.”