The daily sports paper states that the Istanbul giants want to include a buy-out clause in Januzaj’s loan deal.
The Belgium international of Kosovan origin struggled for playing time under United boss Louis van Gaal last season.
Fenerbahçe will take on Shakhtar Donetsk at the end of the month in the first Champions League play-off round and could end up facing United should they progress to the final round.
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.