Moyes knows Januzaj from his time in charge of United and did give the Belgium international playing time during his time in charge.
The 21-year-old did however, struggle last season spending the second half of the league campaign on loan at Borussia Dortmund where he failed to settle in.
Januzaj is reported to be unhappy with his current role at United after being sent to train with the reserves.
Besiktas loaned out star winger Gokhan Tore to West Ham over the summer and have been in the market for a replacement.
Januzaj could also be able to qualify for domestic status due to having part Turkish ancestry.
The United youngster has fallen out of favour with recently appointed manager Jose Mourinho and has been instructed to find a new club according to the Telegraph,
Per the source, United would accept a £10 million offer but both Besiktas and Sunderland are interested in a temporary move.