The Lyon star is wanted by BVB to potentially replace Jadon Sancho, who is most likely to join United.
The Red Devils sold Depay to Lyon in 2017, and included a clause where they have the chance to match any bid from another team to sign Depay.
But Lyon hope to keep the Dutchman and convince him to sign a new contract.
"Manchester [United] has a priority if several clubs line up to buy him," Lyon's director of football Juninho told the club's official website earlier this year.
"But we have not received any offers so far. He is really happy right now in Lyon. He deserves the armband like others.
"Rudi (Garcia) will soon make his decision. He is a technical leader. We will discuss an extension of contract by the end of the year."