The report claims that the Lions made a last-ditch offer for the experienced midfielder on Monday 31 August but were unable to push a deal through before the end of the transfer window.
Galatasaray made several offers for Flamini over the summer but were unable to reach an agreement with the player.
The 31-year-old has just one-year remaining on his current and Galatasaray could sign him on a free-contract in January if Arsenal do not renew terms.
The Lions were desperate to sing a replacement defensive midfielder after losing Felipe Melo to Inter on transfer deadline day.
The Istanbul giants also loaned out Blerim Dzemaili, sold Yekta Kurtuluş and Hamit Altıntop is injured leaving Galatasaray lacking quality and depth in the defensive midfield position.
Galatasaray will either deploy summer signing Jem Karacan or Jose Rodriguez in Melo’s former role.
The Lions will compete in the Champions League group A this season against Benfica, Atletico Madrid and Astana.