According to Gazzetta dello Sport, the 25-year-old’s remarkable turnaround in form has seen him earn interest from many of Europe’s elite clubs.
Kean is currently enjoying the most prolific season of his career, scoring 21 goals and providing three assists in 36 games across all competitions for Fiorentina.
Barcelona are looking to bolster their attacking line up and replace aging striker Robert Lewandowski despite his impressive form this season.
It’s understood Kean has a release clause in his contract and will be available for €52m (£44.2m) between 1 and 15 July this summer.
Fiorentina owner Rocco B. Commisso has insisted the club will do everything they can to keep their top scorer but was unable to rule out a sale.
"I want him to stay here, he's very important to us," he recently told SkySport24. "If someone makes an offer then we'll evaluate, but I'll try to keep him at Fiorentina."