The newspaper Bugun has stated that Galatasaray are interested in opening talks with the 32-year-old and will open negotiations following the end of the season.
Egemen can enter direct talks with other clubs due to his contract set to expire at the end of the season.
The Turkish international has previously also played for rivals Beşiktaş and Trabzonspor. Egemen would join Sergen Yalçın and Burak Yılmaz as the only players to have played for the Süper Lig’s big four (Beşiktaş, Fenerbahçe, Galatasaray, Trabzonspor).
Egemen played an important role in the Fenerbahçe side that won the Süper Lig title last season and was also part of the side that reached the UEFA Europa League semi-final in 2013.