The former Everton attacker revived his career on spectacular fashion this season in Florence, after he had only made the move last summer last summer.
Kean has scored 23 goals in 41 performances in all competitions so far this period.
And that form has brought him to the attention of various large clubs in Europe, including United, Foot Mercato reports.
A move to Old Trafford would mean a return to the northwest for Kean, who spent four years in the books in Everton between 2019-23, although at that time he had loan spells with PSG and Juve.
Kean's contract in Fiorentina has a buy-out clause of € 50 million, which can be activated between 1-15 July.