The Colombian international lost his place in the starting line-up following a dispute with Slaven Bilic and could now be on his way out of the club.
According to the report, Manchester City scouts have been monitoring the 24-year-old in recent months and could launch a bid during the summer transfer window.
City are believed to be in the market for a new centre-back this summer following a string of disappointing performances by Eliaquim Mangala.
Manchester City will also monitor Franco’s performances for Colombia during the Copa America.
Franco joined besiktas in July 2013 for €1.5 from the Colombian club Millonarios FC. The 24-year-old has made a total of 37 appearances for Besiktas in all competitions this season.
The Black Eagles finished the league in third place and will play in the Europa League group stages next season.