Antonio Conte is keen to strengthen his Premier League-winning side this summer and the club's lack of activity is understood to have frustrated the former Italy head coach, who is still to sign a new deal at Stamford Bridge.
In order to appease Conte, Chelsea have stepped up their efforts to add established names to their squad.
The Guardian report that the Blues have already seen one bid for Brazil international Sandro rejected, but have now readied an improved offer worth £55m ($69m).
That latest offer would appear to be enough to trigger a move, although Juventus insist 26-year-old Sandro is yet to tell them he wishes to leave the club.
"A substantial offer has arrived for Alex Sandro," Juventus director general Beppe Marotta said. "We have no intention of selling anyone, but if a player decides to leave, as has happened in previous years, then at the end of the day he has to go, because we don't want to keep anyone against their will.
"But at the moment, there are no such situations and I hope nobody will leave."
Conte is keen to ensure there is greater depth in quality across his side ahead of the 2017-18 season, with Monaco midfielder Tiemoue Bakayoko also on their radar. The Guardian report Chelsea are confident of signing the France international for £40m. He is a player they apparently see as a potential upgrade on Nemanja Matic, who has been linked with a reunion with Jose Mourinho at Manchester United.
With Diego Costa no longer part of Conte's plans in west London, Chelsea remain strongly linked with Everton's Romelu Lukaku. The Belgian could command a fee of £100m if the champions wish to bring him back to the club.