Shaw has clashed with the United boss on numerous occasions since the latter's arrival at Old Trafford, often being overlooked for selection and taken off early when he is chosen in the starting lineup.
Mourinho is understood to be in the market for defensive reinforcements this summer, with left-back one of his priorities after being forced to use Ashley Young in that position for much of the 2017-18 season.
However, The Mail reports that 22-year-old Shaw is willing to see out his contract at the club, which expires next summer after United activated a 12-month extension.
The likes of Tottenham Hotspur, Chelsea, Everton and Monaco have all been linked with a move for the England international, but he is thought to want to wait until at least January to consider his options.
Shaw made just 19 appearances for United across all competitions last season, including only eight Premier League starts.