Kean was born to Ivorian parents in Italy on February 28, 2000 and joined an elite group of 16-year-old Serie A debutants when he replaced Mario Mandzukic at Pescara last weekend.
In Seville on Tuesday Kean showcased his talents to a global audience, when he made a late cameo in a 3-1 win over Sevilla that secured Juve's ticket to the last 16.
The Italian champions are currently ravaged by injuries at a busy period and Mandzukic is the club's only fully-fit striker.
And now Allegri has hinted Kean could make his full Serie A debut away to Genoa on Sunday.
Juventus forwards Paulo Dybala and Mark Pjaca are still recovering from injury while Gonzalo Higuain was left on the bench Tuesday as he recovers from a thigh knock.
"At the moment Pjaca and Dybala are still sidelined, so he (Kean) will be staying with the squad for the moment," Allegri said after the game.
"He's making progress. He's had a few ups and downs, like all young players, but he's on his way back after spending a little too much mental energy."
Kean, a tall and powerfully-built striker who has been compared to Mario Balotelli and has the same agent in Mino Raiola, made such an impact at youth level that he was regularly moved up the age groups.
While videos of his skills have gone viral in Italy, a clip showing his older brother has been featured regularly on sports websites in the past week.
As Kean walked on to make his Juve debut in Turin, his brother wept with emotion as he watched from the stands in Turin surrounded by applauding fans.
Kean did enough in that short appearance to convince Allegri he could be useful on the trip to Sevilla.
With the match delicately poised at 1-1 and Sevilla down to 10 men, Kean replaced Bosnia international midfielder Miralem Pjanic with six minutes to go and was immediately in the thick of the action.
As Juve pressed, Kean rose to challenge for a header that was cleared by a Sevilla head. It fell to Leonardo Bonucci, who slammed a deflected shot past 'keeper Sergio Rico to give Juve a timely 2-1 lead.
Kean was on hand to celebrate and moments later Mandzukic hit a late third for Juve as the Italians secured their place in the last 16.
Kean is not the youngest player to grace Serie A or the Champions League.
He made his top flight debut in Italy at the age of 16 and 265 days and was nearly a year older than Amedeo Amadei when he made his Roma debut aged 15 years, 280 days in May 1937.
The youngest player to play in the Champions League is Nigerian left-back Celestine Babayaro, who started for Anderlecht in a 1-1 draw with Steaua Bucharest aged 16 and 87 days in the 1994/95 group stage.
The youngest scorer in the Champions League remains Peter Ofori-Quaye, who 17 years and 195 days when he scored a late consolation for Olympiakos in a 5-1 defeat to Norwegian side Rosenborg in the 1997/98 group stage.