Bergwijn, 21, is one of the most sought-after wingers in Europe and United are one of a number of clubs said to be interested in signing him this summer.
And Bild claim Bayern are the latest club to declare their interest as they go in search of replacements for Arjen Robben and Franck Ribery.
Manchester City star Leroy Sane is understood to be their top target but their efforts to take him back to Germany have so far come to nothing.
The 23-year-old would cost around £90million and Bergwijn could represent a slightly cheaper alternative with previous reports valuing him at closer to £40m.
He scored 14 league goals in 33 appearances last season and chipped in with another 12 assists for his PSV team-mates.
The report from Bild adds that United could face further competition to sign him from the likes of Tottenham, Ajax and Inter Milan.
Bergwijn was recently asked about moving to Old Trafford and refused to rule it out despite insisting he was happy in Holland.
“Manchester United are a good club, without a doubt,” he said.
“But I am only 21 years old and in no hurry to leave PSV.
“When you join a new club, a better team, you automatically raise your level because of the level of the other players on the pitch.
“If something good comes up, I’ll be ready for it. Otherwise I’ll be happy to stay at PSV.”
United have already signed one winger this summer after paying Swansea £20m to take Daniel James, 21, to Old Trafford.
Bringing in a centre midfielder remains manager Ole Gunnar Solskjaer’s highest priority and they are closing in on Sporting Lisbon’s Bruno Fernandes.
Fernandes, 24, is said to be keen on the move and is just waiting for the two clubs to agree on a transfer fee.
Lazio star Sergej Milnokovic-Savic, 24, is another target and chief executive officer Ed Woodward is reported to have held talks with his agent in London.
It was also claimed that Lazio are desperate to get him off their books after an underwhelming 2018-19 campaign.
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