Elneny is currently on loan at Besiktas and the 27-year-old said he often speaks with Ozil who appears interested in a move to Turkey.
Elneny told Bein Sports Turkey, "Mesut really loves Fenerbahce. He's always speaking about the club and the fans.
"If you ask me, I think he could come to Fenerbahce.
"He's always trying to annoy me by singing Fenerbahce chants."