The Turkish sports channel quote Mundo Deportivo as their source and the Spanish newspaper claim that Van Persie will cost in the region of €3.5 million over the winter break.
The report claims that Barcelona technical secretary Robert Fernandez was at Czech Republic v Netherlands Euro 2016 clash that saw the Dutch lose 3-2 and bow out of the tournament.
Barcelona do not wish to splash out over the winter transfer window but want to bring in experience.
Van Persie only joined Fenerbahçe over the summer from Manchester United on a three-year deal. The experienced striker has four goals and one assist in all competitions all season but is reported to be unhappy with head coach Vitor Pereira due to not starting for the Yellow Canaries over the last few games.
The Dutch star has been playing as a back up to Brazilian keeper Fernandao who already has eight goals this season.