Lewandowski's future at the Allianz Arena has come under question in recent weeks after he changed his agent, appointing Pini Zahavi in a move which increased speculation that he could look to leave the German champions.
A number of Europe's top clubs, including Real Madrid and Chelsea, were immediately linked with the 29-year-old, but German publication Kicker reports that Manchester United are currently the frontrunners for his signature.
The report claims that United are the only club yet to make moves towards signing the Poland international, despite Jose Mourinho previously ruling out the prospect of further attacking additions this summer.
Bayern are reluctant to lose their star striker, though, and would likely command a hefty fee before entertaining an offer, with Lewandowski contracted to the Bavarian outfit until 2021.
Lewandowski has scored 142 goals in just 182 appearances for Bayern, including 32 in 35 outings across all competitions this season.