The Daily Star says United are plotting a January swoop for £25m-rated Shaw.
United boss David Moyes has conceded defeat in his efforts to sign Leighton Baines and is now targeting the Southampton wonderkid to solve his search for a left-back.
However, pulling off a deal won't be easy. As well as the huge fee, Southampton don't need to sell, especially after Shaw signed a five-year deal when he turned 18 in July.
If Saints decide to do business, United won't be the only interested party.
Chelsea are long-term admirers and understood to be Blues fan Shaw's preferred destination.