The 27-year old is a free agent following his release from Paris Saint-Germain last month, but he initially opted against joining a new club.
However, as per reports from Mundo Deportivo, Fenerbahce have now reached out to his representatives over a possible move to the Super Lig, for the second half of the season.
Jese has enjoyed a nomadic career following his exit for the French capital from the Estadio Santiago Bernabeu in 2016.
After failing to establish himself as a regular at the Parc de Princes he has also spent time on loan at Las Palmas, Real Betis and Stoke City and Sporting Lisbon.
However, after a string of unconvincing performances at each of those clubs, he remained a forgotten man on PSG’s books prior to his release by former head coach Thomas Tuchel.