Mourinho had been keen on signing a first choice full-back last summer and had identified Tottenham’s Danny Rose as his preferred target.
Shaw, 22, was recovering from a foot operation at the time and the defender didn’t start for the club this season until early December.
However, Shaw has started in six of United’s last eight matches and is set for an extended run in the side with Ashley Young suspsended.
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The defender’s impressed with his energy in attack but Mourinho is said to have been even more impressed with his renewed focus on his defensive duties.
According to the Mirror, Mourinho took Shaw to one side at the team hotel recently to get assurances over the defender’s fitness and to see if he could deal with the hectic festive schedule.
United have since cooled their interest in Rose and other targets such as Juventus defender Alex Sandro.