The Blues made a first approach for the centre-back this month when representative Fali Ramadani met with the Bianconeri's sporting director, Fabio Paratici , in Milan.
The Serie A champions knocked back that initial £36 million ($47m) offer, expecting around £10m more for the 23-year-old.
And the London club are willing to match that price in their next offer to close in on one of their top targets for the summer.
Chelsea are willing to offer Rugani a contract worth around £4m per season to reunite him with Maurizio Sarri.
The new coach worked with Rugani for two years at Empoli when the defender was sent on loan from Juve.
Rugani played a pivotal role in the defence as Sarri's side gained promotion from Serie B. He then started every game in the top flight the following season.
And the admiration is mutual, as Rugani recently said of Sarri: "He is a person whom I respect a lot, who gave me a lot and who taught me many things."
Since his return to Juventus, Rugani, who has three years left on his contract in Turin, has been unable to nail down a spot in Massimiliano Allegri's first XI, but started 22 matches in the league last season.
Although a move to the Premier League looks to be on the cards, the player travelled with his current team-mates to the United States on Monday as they begin their pre-season tour, in which they will face Bayern Munich, Benfica and Real Madrid.
As well as Rugani, Chelsea are confident of landing Aleksandr Golovin from CSKA Moscow, whose former team-mate seemed to confirm a move had been completed over the weekend .