The Brazilian has undergone a medical at the Serie A club Fiorentina who posted a photo of the 25-year-old on their Twitter feed.
"We've not been able to get him as many games as he'd hoped, or we hoped we might get him, so we feel it's a good opportunity for him," Moyes told a news conference.
"He's going to a good club in a good league, so hopefully it gets him some benefit."
Brazil international Anderson has made 179 appearances for United since signing from Porto in 2007 in a combined deal with winger Nani worth £30 million.
He has started only three games under Moyes this season and his time at United has been blighted by inconsistency as he has failed to tie down a first-team place and showed only glimpses of his potential.
He also struggled with injury and fitness problems during his time at Old Trafford.
Anderson's departure could prompt Moyes to make a move in the transfer market, with United fans crying out for reinforcements in midfield. Marouane Fellaini was their only arrival in the close-season.
Fiorentina are fourth in Serie A, 15 points adrift of leaders Juventus.