According to Sport1 (via The Mirror), United have made contact with Bayern over a possible transfer of the German centre-back.
Ole Gunnar Solskjaer is believed to have made it clear to Ed Woodward that he doesn't want a repeat of last summer, where United failed to sign a defender.
United have been linked to a host of Europe's top centre-backs, and Süle is the latest to have been approached.
Bayern signed Süle for £21m from Hoffenheim two years ago, but the 23-year-old has three years remaining on his deal, meaning it will take a substantial bid to prise him away.
A starting offer of about £40m is reported to be a sensible starting point.
Kalidou Koulibaly, who has been linked to United this summer, is still unsure about his future, while Mathjiis de Ligt of Ajax is reportedly close to agreeing a move to Barcelona.
With the market shrinking somewhat, United are keen to make progress in their hunt for defenders.
Get the best features, fun and footballing frolics straight to your inbox every week.