The striker has enjoyed yet another prolific campaign in Germany, scoring 38 goals in all competitions.
Lewandowski’s current contract expires in 2021 and Bayern are keen to extend his deal by another year.
According to Kicker, the club will open talks following their German Cup final clash against RB Leipzig on May 25.
However, the report claims Lewandowski is considering his options and could look to quit the club this summer.
And both United and PSG have shown interest in snapping him up.
Ole Gunnar Solskjaer is desperate to strengthen his squad after guiding the Red Devils to a sixth-place finish in the Premier League.
United are reportedly keen to stray away from their model of bringing in marquee signings and would prefer to develop young talent.
The likes of Aaron Wan-Bissaka, Declan Rice and Dan James have all been touted, while Jadon Sancho remains their priority signing.
“#ManchesterUnited insist to sign Jadon #Sancho.”
Italian journalist Nicolo Schira
However, with Champions League football in their sights, Lewandowski’s availability may force them to rethink their strategy.
Romelu Lukaku has been linked with a move to Juventus after his agent made it clear he was rethinking his future.
Solskjaer wants competition for Marcus Rashford up front, and Lyon striker Moussa Dembele has been scouted.
The Frenchman would likely be more cost effective, however Lewandowski is more prolific.
And the Poland international might not be the only big-name arrival at Old Trafford.
Italian journalist Nicolo Schira tweeted: “#ManchesterUnited insist to sign Jadon #Sancho.
“Yesterday new contacts between Borussia Dortmund).
“#RedDevils are also interested in Philippe #Coutinho (with the arrival of Griezmann he can leave from Barcellona) and Youri #Tielemans. #transfers #MUFC #mutd.”