It is reported that Chelsea, Milan’s rivals for the signing of Bernard, have told the player he has until Tuesday to decide whether he wants to join the club.
The Rossoneri and the London side are both keen to reach an agreement with the Brazilian, who is a free-agent after leaving Shakhtar Donetsk.
And Gazzetta dello Sport reports the Blues have issued the 25-year-old with a deadline to make his decision.
The Stamford Bridge club, now coached by Maurizio Sarri, have offered Bernard a contract as they look to bolster their squad for the new season.
But they are set to pursue other targets if the winger has not accepted the proposal by the middle of next week.
Bernard spent five seasons with Shakhtar before his contract with the Ukrainian side expired this summer.