The report claims that the Lions will attempt to sign the youngster on a temporary move as a change in transfer policy.
The Istanbul giants are currently having to change their spending structure after being hit with a one-year European competition ban for failing to meet FFP standards.
Galatasaray will be aiming to reduce their transfer fee payments over the summer and target free agents as well as loan moves.
The Lions had success bringing in Manchester City defender Jason Denayer last season who featured as a first team player and established himself as one of the best performing defensive players of the season.
This is not the first time Galatasaray have targeted Januzaj they were also closely linked with the Belgium intentional last summer as well.
Januzaj struggled for playing time under the command of Louis van Gaal last season making just five Premier League appearances.
The 21-year-old did spend the first half of the season out on loan at Borussia Dortmund which suggests that United may be keen on letting the 21-year-old leave on a temporary move to gain experience and develop.