According to the Mirror's Adrian Kajumba, Juventus are aware of the Blues' interest and discussed the issue with Rugani's camp on Thursday during a meeting in Milan between club officials and his representatives.
Juventus bought the centre-back while Chelsea boss Antonio Conte was manager in Turin, and the coach also called him up to Italy's senior side for the first time, although Rugani did not make his Azzurri debut until after Conte had left for Stamford Bridge in 2016.
Maurizio Sarri—who is said to be the primary candidate to replace Conte this summer if he leaves the Blues—oversaw Rugani's formative years at Empoli, so he too is familiar with the player.
The 23-year-old only made 26 appearances for the Bianconeri last season, but with Andrea Barzagli now 37 and Giorgio Chiellini turning 34 in August, he's in line to become a key player for the Bianconeri in the coming years.
Football writer Adam Digby has been critical of Rugani and believes he needs to improve if he's to reach his potential, though Bleacher Report's Gianni Verschueren feels the defender has done well:
Rugani reads the game well and positions himself accordingly, so he's not a defender you will regularly see making last-ditch challenges.
Copa90 US' Aaron West was impressed with the defender handled Andrea Belotti when Juventus beat local rivals Torino in February:
He's technically adept and composed on the ball, and he has a bright future ahead.
With the likes of Antonio Rudiger, Andreas Christensen and Cesar Azpilicueta available at centre-back along with Gary Cahill, David Luiz and Kurt Zouma, Chelsea aren't in desperate need of reinforcements in that position, though Rugani would arguably be an upgrade on some of them.
He has been groomed to play a key role at Juve, though, so it's difficult to imagine them letting him go easily.