The 21-year-old can play on both the left and the right, and has been liked with a number of elite European clubs.
Chiesa is unlikely be sold in January, with any transfer likely to come in the summer.
Fiorentina value the youngster at up to €70 million (£62m/$79m) after he became an indispensable part of Fiorentina's XI in Serie A.
Chiesa has already made 85 appearances for the Viola, scoring 12 goals and provided 17 assists, while he has also won 11 caps for Italy.
Juve have previously shopped at Fiorentina, snapping up Federico Bernardeschi in 2017 for a fee of £35m.
In June, Chiesa's father claimed that he would remain at the club for the duration of the season, but it remains to be seen if Juventus' interest will change his mind.
"Now he has to think about the national team, because it represents a great satisfaction for him," Enrico Chiesa told reporters.
"For the rest [transfer rumours], he has a contract and he is very happy. Now he focuses on the national team and then from Monday think about enjoying a month off, because he deserves it.
"It was a very hard season. He and his team-mates still have to overcome the [Davide] Astori drama [after his death in March].
"The next few weeks will serve him to rest and then get ready again."
The two clubs also did business regarding Marko Pjaca, with Fiorentina taking the winger on loan in the summer.
Juve received €2 million (£1.8m/$2.3m) as a result and, while Fiorentina have the option to make the move permanent for €20m (£18m/$23m) next summer, a deal remains in doubt with Pjaca failing to impress in Florence so far.