United have been linked for some time with a move for Lewandowski, who has forged a formidable reputation with Dortmund as the German side claimed back-to-back Bundesliga titles in the last two seasons.
After leading the line and claiming 22 goals in 34 games last time round, The 24-year-old striker has scored 10 goals in 16 appearances this season.
The Red Devils have an embarrassment of riches in attack after adding Robin van Persie over the summer to play alongside the likes of Javier Hernandez, Danny Welbeck and Wayne Rooney.
Despite that, Sir Alex Ferguson is said to be close to landing Lewandowski, who was a reported target for Manchester United before they eventually opted to pursue Van Persie.
Bild claim that Manchester United have made a 'mega offer' for the Polish international and are prepared to double his current €2.5 million wage packet in order to take him to Old Trafford.
The paper report that Lewandowski is close to accepting the offer and could make the move to Manchester in the summer.
Recently Dortmund have been forced to admit Lewandowski will likely leave the club in the summer after turning down the offer to extend his contract beyond its current 2014 end date - although they insist that no offers have been received for the striker as yet.