Last weekend, Shaw was substituted at half time of United's FA Cup tie with Brighton & Hove Albion and his performance was openly criticised by manager Jose Mourinho.
The Portuguese's comments have led to reports suggesting that Shaw could leave Old Trafford at the end of the season, and a wide range of Premier League clubs have been linked with the England international's signature.
According to ESPN, Everton are weighing up the possibility of making an approach for the left-back with the Toffees requiring a long-term replacement for Leighton Baines.
Shaw has made just 15 appearances for United in all competitions this season, but Everton boss Sam Allardyce would likely regard the 22-year-old as first choice on the left side of defence.
The player has just 15 months remaining on his current contract.