Juventus had a bid of £50 million turned down by Fiorentina over the summer but are expected to return with a higher offer once the window reopens, per the report.
Chiesa has come through the ranks at Fiorentina and emerged as one of Italy's brightest young talents.
He is already an established international and made an impression during Italy's 2018-19 UEFA Nations League games against Poland and Portugal.
Chiesa came on as a substitute against Poland and caused problems with his pace. He won the penalty from which Jorginho scored Italy's equaliser.
Football writer Peter Galindo said he could be a regular for Italy soon:
His performance against Portugal was more frustrating, as he often opted to shoot when he would have been better off crossing for a team-mate.
Football writer Adam Digby shared his view:
Chiesa has also started the season well with Fiorentina. He has a goal and an assist in his first three league appearances.
There's no doubting he has huge potential, and Fiorentina may struggle to keep hold of him if some of Europe's big spenders come calling.
Chiesa has said he made the right choice staying in Florence this season after interest from Juventus, per Corriere della Sera (h/t Fissler):
"The summer has been peaceful, there were no anxieties related to the market because, in agreement with the club, I decided that I would stay in Florence. It was the best choice, and knowing that I'm worth so much does not detract from my sleep, nor does it affect my life. Prices in recent years have skyrocketed, and I'm simply going to try and not think about it."
If Chiesa can continue his progression, his price tag is likely to rise even further, which could price Napoli out of a move. Juventus will not be afraid to spend big, but Fiorentina may prefer to sell to Chelsea rather than a Serie A rival.