Shaw, the latest in a long line of young prospects to emerge from Southampton's youth academy, has impressed for the south coast club already this season.
The 18-year-old featured for the Saints in their 1-1 draw against Manchester United at Old Trafford on Saturday.
Now according to the Times, Manchester United boss David Moyes has identified Shaw as a target, and is reportedly 'increasingly likely' to move for him in the January transfer window.
The Red Devils tried and failed with repeated attempts to sign Leighton Baines during the summer transfer window and are said to now be eyeing Shaw as a more attainable target, although the Everton man remains on their radar.
Moyes also failed with a last-ditch attempt to take Real Madrid full-back Fabio Coentrao on loan in the last minutes of deadline day.
Tottenham and Chelsea have also been linked with the England U21 defender, who put pen to paper on a new long-term deal in the summer.
The Manchester United boss is under intense pressure to turn things around after a slow start to his first season in the Old Trafford hot seat.
Not helped by a summer transfer window in which the reigning Premier League champions missed out on a host of high-profile transfer targets, Moyes is said to be eyeing the winter window already as the solution to his problems so far.
The Times say Porto man Eliaquim Mangala is being monitored, while the Scot is also keen on two midfielders and a new winger.