The two London clubs have both been linked with moves for the £20million-rated Barcelona attacking midfielder.
But, according to Fanatik, they will face stiff competition from his homeland for his signature this summer with Besiktas lining up a move.
It has been suggested that the Black Eagles have already enquired about his availability and are hoping to complete a season-long loan deal for Turan.
That idea could appeal to Barca, who only landed the 29-year-old 12 months ago from Atletico Madrid.
Turan had to wait until January to make his debut due to the Spanish club’s transfer ban and managed just nine starts in La Liga in the second half of last season.
It has been suggested that Barca could allow him to leave this summer but Turan is said to be happy at the Nou Camp and is willing to stay and fight for his place.
But that will not put Besiktas off bidding for the former Galatasaray ace as they strive to strengthen their squad during the summer.