The former-Porto star has struggled for playing time in recent seasons, first under Sir Alex Ferguson and latterly under Moyes.
He started against West Brom on Saturday - just his third Premier League appearance of the campaign - but was largely ineffective and was replaced after 68 minutes.
The Brazil international was a big-money recruit from Porto back in 2007, but he has failed to establish himself as a first-choice midfielder at Old Trafford. Injuries have not helped, as Anderson has struggled to stay fit for extended periods of time.
And the report claims, having been left at home for the Champions League trip to Shakhtar Donetsk, Anderson could be set to leave United sooner rather than later.
Earlier this year he admitted he's tried to exit the club on several occasions, only for the United hierarchy to convince him to stay.
At that stage, the likes of Porto and Benfica, as well as Brazilian clubs Sao Paulo and Gremio, were linked with the United midfielder.
But with new management now in charge and Marouane Fellaini blocking his path into the starting eleven, the Brazilian could be allowed to leave - either in January or next summer.