Manuel Pellegrini is considering signing a new left back in the summer, with Gael Clichy having lost his first team place, and concerns remaining over the defensive capabilities of Alexsander Kolarov.
At just 18-years-old, the Southampton youngster represents an attractive prospect of a long term replacement.
Chelsea are still favourites for his signature, with Shaw understood to have been a Blues fan as a child.
The club’s training ground is located in Cobham, where the full-back’s parents live.
Manchester United are still keen on signing Shaw, even if Patrice Evra decides to sign an extension to his contract which expires at the end of the season.
The reigning Premier League champions only have an outside chance of completing the deal however, especially since they are looking increasingly unlikely to be able to offer Champions League football.
Evra and United hold an option that would allow the Frenchman to extend his contract at Old Trafford by a year.
Whether Evra would be willing to drop down to second choice remains to be seen, with Moyes intent on signing a new left back whether the 32-year-old decides to stay or not.