The report claims that the Yellow Canaries have made an enquiry for the Belgium international but will only make an offer should they progress past the preliminary round of the Champions League play-off stage against Shakhtar Donetsk.
Fenerbahçe will take on Shakhtar at the end of the month and could end up facing United should they book a place in the final play-off 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.
The Belgium international of Kosovan origin struggled for playing time under United boss Louis van Gaal last season making just 18 Premier League appearances.