The teenage sensation earned rave reviews during his breakthrough season with Southampton in 2012/13, and his impressive performances have attracted admiring glances from a host of the Premier League's biggest clubs.
Shaw is regarded as one of the hottest young properties in all of Europe, and the fact he is yet to sign a senior contract with the south coast club means there is a strong chance he could leave St Mary's this summer.
The England U21 international is rated at around £10million - a valuation that is well within the means of both Chelsea and Manchester United, if they choose to ramp up their interest when the transfer window reopens on July 1.
Both clubs are well-stocked in the left-back area, with Chelsea already possessing Ashley Cole and Ryan Bertrand, while Manchester United can call upon Patrice Evra and Alex Buttner - whom they signed from Vitesse Arnhem last summer.
However, Moyes is understood to have prioritised a new full-back as one of his first areas to strengthen, while Jose Mourinho is also looking towards the long-term future at Stamford Bridge, with Ashley Cole entering the twilight of his football career.
Manchester United will be determined not to miss out on self-proclaimed Chelsea fan Shaw, though, after they saw Gareth Bale snub them in favour of a move to London, when he joined Tottenham Hotspur from Southampton a few years ago.