Sky Italia, who broke the news that coach Josep Guardiola was on his way to Bavaria in December, claim that the Polish striker will move between the Bundesliga rivals at the end of the season.
Guardiola has apparently given his backing to the deal as he looks to impose his own style of play on Bayern, who are clear at the top of the table this season and preparing to face Arsenal in the Champions League.
United were strongly linked to the 24-year-old hitman, who scored 43 goals in 86 league games as Dortmund won back-to-back titles in 2011 and 2012.
But Borussia resisted attempts to sign Lewandowski shortly after Shinji Kagawa made the move to Old Trafford and Sir Alex Ferguson then concentrated his full effort on signing Robin van Persie.
However, Lewandowski has refused to sign a new deal this season and with less than 18 months to run on his contract, Borussia officials are aware they may have to cash in on him before he enters the final year of his current agreement.
Dortmund chief executive Hans-Joachim Watzke said yesterday: "There is no new stance. But our aim is of course still to extend with Robert."
Lewandowski stonewalled questions about his future yesterday, saying he will definitely stay at the club until the end of the season and that he was staying focused on the run-in.