For 26-year-old Alex Sandro, Chelsea will up their bid from £55.2m to £58m and have already told the player that he will earn more money if he agrees to make the move to Stamford Bridge.
However, Juventus are keen to do all they can to resist the Premier League champions and are offering the 26-year-old a contract until 2022, with a pay rise of their own to counter the approach.
Massimiliano Allegri's side don't want to sell the wing-back but Chelsea believe they could be tempted by a big price.
Juventus are more open to letting go of Bonucci, who Conte is equally keen on bringing into his back line.
The ex-Bianconeri boss wants Chelsea to make a big offer for the 30-year-old centre-back and Juve are willing to consider a sale, if the price is right, after a season of turmoil off the pitch. Bonucci recently denied a bust-up with Paulo Dybala at half-time of the Champions League final but his relationship is known to be strained with Allegri.
Chelsea have yet to make an offer for Bonucci and he is older than the normal profile of player that the club likes to sign, but Conte has made his intentions clear to the board after verbally agreeing to stay until 2021 with a pay-rise of his own.
Conte is willing to offload Nathan Ake or Kurt Zouma to make room for Bonucci, either on a temporary or permanent basis.