According to Talksport Mahrez has sparked interest from the La Liga champions as well as from Arsenal followng a stunning start to the season.
Mahrez has scored four goals in four Premier League appearances this season and also played a major role in keeping Leicester in the Premier League last season.
Barcelona are on the hunt for a new winger following the departure of Pedro to Chelsea in the summer.
Turan is unable to play for Barcelona due to a FIFA imposed transfer ban however, the 28-year-old played an hour of football against the Dutch on Sunday. Barcelona will be able to register players once again in January 2016.
The 28-year-old has now scored twice and provided two assists in his last three Euro 2016 Qualifying games for Turkey.
Turkey take on Iceland and the Czech Republic in their next two Euro 2016 qualification games.