United are in desperate need of a left-back, with Luke Shaw out of favour and converted winger Ashley young currently filling in more often than not.
Tierney is considered to be one of the most promising young defenders in the British Isles and has already racked up 120 professional appearances by the age of just 20.
The youngster is thought to tick a lot of boxes for Mourinho, particularly in terms of his leadership skills and versatility, with Tierney having already captained his country and shown a willingness and adeptness to fill in anywhere across the back four.
Scottish publication 'The Daily Record' reports that Tierney is one of Mourinho's top tagets as he looks to complete a full squad overhaul and that a fee of £20m could be enough to do a deal.
That fee means that he would come considerably cheaper than other alternatives, such as Tottenham's Dnny Rose.
Tierney signed a six-year contract at Celtic Park just last usmmer, but there is an acceptance in Glasgow that when the big boys come calling, it will be impossible to keep the starlet at the club.