The Red Devils will have to cope without Nemanja Vidic from the end of the season, with the rugged Serbian set to leave the club once his contract expires.
David Moyes is desperate to land a replacement for his influential star, and is believed to have settled on Savic.
The Montenegro international previously enjoyed a brief and rather unsuccessful spell with City before leaving in 2012, but has since thrived at Fiorentina.
United are said to have been impressed by displays, and are confident that they can lure him back to England to prove himself to his former employers.
It is understood that Moyes talked through a possible summer move with the Italian club when they loaned them Anderson in January, but it remains unclear as to whether a deal will be workable.
Savic is a key member of La Viola’s first-team, and has been praised for his composed playing style.
United are also believed to be tracking Eliaquim Mangala of Porto and Benfica’s Ezequiel Garay.