The Red Devils have been heavily linked with the Polish international in recent months, with manager Sir Alex Ferguson believed to be highly interested in a summer deal.
However, reports have now emerged that Pep Guardiola will make the front-man his first signing at the club when he takes over at the end of the season.
The 24-year-old has netted 17 goals in all competitions this term, alerting their Bundesliga rivals.
Bayern are thought to be keen to reinstate themselves as Germany’s top club side after surrendering the last two titles to Dortmund.
A move for Lewandowski would serve to strengthen their already fearsome attacking line, whilst weakening their main rivals ahead of next season.
Fergie was believed to be close to bringing the striker to Old Trafford last month, but negotiations between United and Dortmund broke down.
Despite having the likes of Robin van Persie, Wayne Rooney, Javier Hernandez and Danny Welbeck available to him as centre-forward options, the Scot was thought to be keen to add the former Lech Poznan man to his armoury.
But, United were confident of concluding a deal, after taking Shinji Kagawa from the German club last summer.