The Poland international's future at the Bundesliga club has fallen into doubt after claims that the 28-year-old is unhappy with the German outfit.
One of the forward's representatives Maik Barthel recently revealed that his client was frustrated that the club and head coach Carlo Ancelotti failed to support him in his attempts to end last season as top scorer.
In the end, Lewandowski netted 30 goals, but Borussia Dortmund's Pierre-Emerick Aubameyang took the individual title after bringing up his tally to 31 on the final weekend of the season.
According to Polish journalist Maciej Iwanski, the Bayern forward is contemplating his future and there has been interest from both Chelsea and United, but a move to London is his preferred choice.
The Daily Star translates Iwanski's tweet as saying: "Today I had a very interesting conversation in the context of Robert Lewandowski with a knowledgeable Spanish colleague. He claims that Real [Madrid] will give up the fight for Robert, because they have Benzema, and he has not harmed Zidane.
"Besides, there is no good offer for the Frenchman [to leave Real]. Also. Really interesting to note when he said that 'Besides, Real is not going to chase with others interested.' Reportedly the wish of Robert has been... Chelsea.
"For years Jose Mourinho has been interested in Lewandowski. The player himself is considering transfer, but to London. [Lewandowski's] agent, Cezary Kucharski, of course, does not confirm, but the one who called me - the information is checked. Believe me."
The striker controversially joined Bayern from Dortmund in 2014.