The 21-year-old has regained his form after joining PSG on loan from Everton last summer, scoring 16 goals across all competitions for the French club.
Having been impressed with Kean, the Ligue 1 giants are now keen to buy the striker from Everton, while Juventus are said to be interested in re-signing him.
There have been suggestions that the Italy international prefers to stay at PSG despite having the chance to return to Juventus and having a contract with Everton until 2024.
However, according to Italian daily Gazzetta dello Sport, Allegri returning as the coach of Juventus could put the Bianconeri in a strong position to sign Kean.
Allegri, who left Juventus at the end of the 2018/19 season, has been linked with a return to the Old Lady after Andrea Pirlo failed to retain the Serie A title this season.
It was under the Italian tactician that Kean made his Serie A and Champions League debuts and a chance to play under his management again could tempt the striker to return to Turin.
However, it remains to be seen if Juventus will sack Pirlo and step up their interest in bringing back Allegri, and whether they can afford to re-sign Kean.