Max, 26, has been seeking a new adventure for some time after consistently being one of the most productive full-backs in the Bundesliga.
According to SPORT BILD, a €7.5m offer was made to Augsburg on August 18, possibly from PSV Eindhoven.
New PSV coach Roger Schmidt knows Max well from his three years in charge at Bayer Leverkusen.
Reuter: We have an offer
FCA’s general manager Stefan Reuter confirmed a bid had arrived, without saying which club it came from.
“It is correct that we have received an offer to buy Philipp Max for the first time,” Reuter said. “We are in contact with the interested club.
“However, we still have differences when it comes to the transfer fee.”
Several clubs have made inquiries for Max, who has spent the past five years at Augsburg.
Max is renowned for his ability to both score and create goals from his left-back position.
Last season he found the net eight times while clocking up six assists in the Bundesliga.