Tierney has become a key player for Scottish champions Celtic since breaking into the first-team in 2014, playing 110 games for the club in all competitions.
Mourinho is keen to sign another left back in January and Tierney has been placed on the shortlist of potential targets, according to the Daily Record.
United, who are currently second in the English top-flight – 15 points behind rivals Manchester City – are also expressing interest in Fulham’s teenage sensation Ryan Sessegnon.
Mourinho is said to have been impressed by Scotland ace Tierney and England youth team star Sessegnon and views both as cheaper options to his preferred target, Juventus’ Alex Sandro.
Despite his young age, Tierney can already boast to have won two Scottish Premiership titles and played an integral part in Celtic’s Champions League campaigns over the past two years.
The two-time PFA Scotland Young Player of the Year has also become a regular for Scotland since making his international debut in 2016.