The Lille forward has attracted the attention of several big clubs after six goals in nine Ligue 1 appearances this season.
Pepe has also bagged six assists in those games, and United and Arsenal are weighing up transfer moves for the 23-year-old.
But SportBild claim Bayern want to bring Pepe to Germany and are desperate to seal the deal in January rather than next summer.
Talks have been scheduled with Pepes team in the coming weeks, with Lille thought to value their star attacker at around £30million.
But should United or Arsenal react to Bayerns interest, that could drive the asking price up.
Everton, Sevilla and Barcelona have also been credited with an interest in Pepe and a bidding war could still erupt for his services.
Pepe is under contract at Lille until 2022 and the French side are under no pressure to sell their prized asset.
Arsenal boss Unai Emery knows Ligue 1 well after two years in charge at Paris Saint-Germain, while Jose Mourinho has made no secret of his desire to sign a right sided winger.
Pepe has played on the right wing for Lille this season, and has helped fire his side to second in the table.