Fiorentina put a photograph on Twitter of the Brazilian, who has made 177 appearances for United since joining from Porto in 2007, being examined.
The 25-year-old has started just three matches since David Moyes succeeded Sir Alex Ferguson as manager last summer.
“We’ve not been able to get him as many games as he’d hoped, or we hoped we might get him,” said Moyes.
“We feel it’s a good opportunity for him.”
Anderson, who cost United a reported fee of about ÂŁ20m, has only completed 90 minutes once since the start of last season, in the 5-5 draw at West Brom that ended the Reds’ title-winning campaign in May.
His last appearance was in the 2-0 Capital One Cup win at Stoke City on 18 December.
Moyes is hoping to strengthen his squad in January, but admitted earlier this month he may struggle to do so because there are not many suitable players available to sign in the mid-season transfer window.
Midfielder Marouane Fellaini, who played under Moyes at former club Everton, was the only new senior arrival at United last summer.
Fiorentina, managed by Vincenzo Montella, are fourth in the Serie A table, 15 points behind leaders Juventus.
Written by Steve Milne – Have your say below 🙂
