The Spanish playmaker only signed for the Red Devils from Athletic Bilbao over the summer in a deal worth close to £30m, but Louis van Gaal is understood to be ready to accept a big offer for the passer from the Bundesliga champions.
Bayern are said to be willing to pay £7m and allow Shaqiri to head in the other direction, which has tempted the Dutch boss.
Herrera has featured just five times since August for United, and has struggled to convince LVG that he’s the right man to be the creative hub of his side.
United themselves are keen on Shaqiri, who starred for Switzerland at the World Cup.
The stocky winger is also wanted by the likes of Liverpool, Tottenham and Juventus.
Van Gaal is said to be keen to get more from his wide options after a tricky start to the season, with the 23-year-old a player he sees as a wise addition.