The 31-year-old has just one year left on his City contract and with talk that Pep Guardiola is looking to bring in three or four full-backs this summer he could well decide that his time at the Etihad has come to an end.
Besiktas are long-term admirers of the Serb, having also tried to sign him last summer.
The 'Daily Mail' claims that the Turkish champions have offered Kolarov a 4-year contract to make the move East.
Kolarov was a regular in the City side last year, making 27 starts in the Premier League, thought it remains to be seen where his future lies.