Per the source, the Black Eagles want the City attacking midfielder on loan and the Premier League outfit are happy to see the French international leave after falling out of favour with head coach Pep Guardiola.
Besiktas manager Senol Gunes recently confirmed that club president Fikret Orman is in England holding talks with City.
Gunes was quoted by Turkish-football as saying: “The boss [Fikret Orman] is in London holding talks, we left transfers late last season as well.”
The Eagles are also in talks with the Citizens over the loan transfer of Wilfried Bony who has also been instructed to find another club.
Nasri has won two Premier League titles and establish himself as a star player at City until the appointment of Pep Guardiola.
Besiktas would be able to offer the experienced attacking midfielder guaranteed first team and Champions Leage football this season if he were to make the move.