The Algerian international is expected to leave Valencia at the end of the season when his contract runs out.
The Turkish newspaper Takvim states that Fenerbahce have offered the 26-year-old €3.2 million per season.
The report goes onto state that Tottenham and Inter are also in the race to sign the in-demand midfielder.
Fenerbahce have already signed eight players this season including Lille defender Simon Kjaer, 2014-15 Süper Lig top scorer Fernandao, Turkish international Şener Özbayraklı and Brazil international Josef De Souza.
Feghouli has four assists and seven goals in 36 appearances for Valencia this season. The 26-year-old has made 39 appearances for the Algerian national team.