The 25-year-old Brazilian, who arrived at Old Trafford from Porto in 2007, has made just four Premier League appearances for David Moyes' men this season.
Having been plagued by injuries for much of his time with United, Anderson now appears set to seek regular first-team action in Serie A.
Anderson arrived in Florence on Thursday afternoon and met Fiorentina's sporting director Daniele Prade and technical director Eduardo Macia.
Fiorentina revealed on their website that the player is set to undergo a medical on Friday, with a view to becoming Vincenzo Montella's latest signing.
Montella signed goalkeeper Antonio Rosati on loan from Serie A rivals Sassuolo for the rest of the season on Thursday.
Rosati has started only two Serie games for Eusebio Di Franceso's side this season, with his last appearance coming back in September.
The 30-year-old passed a medical on Thursday and has swapped a relegation battle for the Florence club's quest to qualify for Europe.
The Tivoli-born keeper will provide competition for Brazilian Neto and is set to face the media at a press conference on Friday.
His arrival follows the loan signing of striker Alessandro Matri from Milan.
