The report states that Sunderland have enquired about the availability of the Belgium international and that Dick Advocaat is keen to sign the young midfielder.
The 20-year-old has struggled for playing time since the appointment of Louis van Gaal at United.
Fenerbahçe 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.
Januzaj would be able to play under domestic status in Turkey due to having part-Turkish heritage.
The Belgium international of Kosovan origin struggled for playing time under United boss Louis van Gaal last season making just 18 Premier League appearances.