Nastasic, 19, has only spent one season at Fiorentina, making 25 appearances as his side finished 13th in Serie A. At international level he plays alongside Manchester United captain Nemanja Vidic.
City's Italian manager Mancini has been tracking the player for some time and is thought to see him as a better prospect than Savic.
Montenegrin Savic, who started the Community Shield tie against Chelsea but has otherwise struggled to make the first team, signed for City from Partisan Belgrade in July 2011 - the summer that Nastasic moved from the same club to Fiorentina.
After a 2-2 draw against Liverpool on Sunday, Mancini said that his side needed to improve their defensive performances.