The Belgium international has also been closely linked with Fenerabahçe over the summer transfer window.
Fanatik claimed that the Yellow Canaries are also targeting a move for the young winger.
The Turkish sports newspaper report that the Super Lig outfit are open to either loaning Januzaj with an option to buy or buying the Belgium international on a permanent deal.
The Istanbul giants 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.
Porto duo; defender Ba and goalkeeper Fabiano also joined the Canaries on loan.
Januzaj would be able to play under domestic status in Turkey due to having part-Turkish heritage.
The 20-year-old of Kosovan origin struggled for playing time under United boss Louis van Gaal last season making just 18 Premier League appearances.