The Brazil international has made 179 appearances for United, but his seven-year association with the club looks set to end after Fiorentina posted a photo on their Twitter account of the 25-year-old undergoing medical checks.
Anderson, who joined from Porto in a combined deal with Nani for 30 million pounds ($49 million), has been on the fringes of the first team this season having started three games under new manager David Moyes.
His time at United has been blighted by inconsistency, with the player unable to tie down a first-team place, showing only glimpses of his potential.
He also struggled with injury and fitness problems during his time at Old Traffotd.
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.
(Reporting by Toby Davis; editing by Ed Osmond)
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