Tosun was loaned to Crystal Palace in January, but returned in March due to a knee injury. The 29-year-old is expected to be fully recovered by November.
However, Sporx says Tosun could return to Turkey before then with Fener willing to sign him as he does his rehab.
A loan deal is being drawn up for his and Everton's consideration.
Fenerbahce see Tosun as a possible replacement for Vedat Muriqi - who is being chased by bigger clubs.