The Daily Record (h/t Jack Otway of the Daily Express) reported a United scout was present to watch the left-back as the Scottish champions played Dundee.
Mourinho is yet to find a regular left-back within his ranks, with Antonio Valencia solidifying the right-back position as his own.
Luke Shaw's lack of fitness has seen him fail to impress Mourinho, prompting the self-proclaimed Special One to assess the transfer market.
French newspaper L’Equipe (h/t Robert Collins of The Scottish Sun) recently rated Tierney as one of the top 50 players in the world under the age of 21. The defender was ranked at No. 37 by the publication.
The Scotland international has continued to impress for the Glasgow giants, and he could develop into a long-term option for United.
Here is the 20-year-old in action:
According to Calciomercato (h/t The Mirror), the 16-year-old is being watched by the Premier League clubs after impressing for Genoa in Serie A. The player nearly joined Inter Milan in a big-money move in the summer before the deal fell through.
Per Alec Shilton of The Sun, Pellegri recently became the youngest player to score a brace in the Italian top-flight.
Here is the player in action:
Shaw's future at Old Trafford appears unstable, and if Mourinho moves for Tierney, it will be the end of the England international's spell at the club.
The player was lauded as he moved from Southampton to the Theatre of Dreams, but injury and poor form have blighted his career.
Shaw will still have opportunities to impress in the coming weeks, but his chances are diminishing with a manager who has little patience as he chases silverware.