Juventus increase Oscar bid by £1.8m
The Turin club are keen to land the Blues' Brazil international, who played 41 games last season.
They reportedly made an £18m offer last month which was rejected by Stamford Bridge bosses and Corriere dello Sport claim a new bid is on the table.
Manager Jose Mourinho is said to be open to a sale of his midfielder but only if the price is right.
Oscar, 23, was recently named in a supposed swap deal involving Juve ace Paul Pogba, who Chelsea want to snap up.
The Blues are thought to have offered Juve £35m plus Oscar for Pogba's services, although Barcelona remain likely to land the ace.
Were Oscar to leave Stamford Bridge, Chelsea would likely begin the hunt for a replacement attacking midfielder.
Mo Salah is also expected to depart to Fiorentina for £12.7m, meaning the club have funds to improve their squad ahead of the new season.