Jose Mourinho is on the lookout for a new left-back with Luke Shaw likely to depart this summer.
Tierney could fill that void, having helped the Hoops win the Scottish Premiership.
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Luke Shaw looks destined to quit Manchester United
And the Manchester Evening News claim the 19-year-old wants to move to United if he does indeed leave Celtic at the end of the season.
But it is said Tierney has made it clear he would rather make the move to Old Trafford.
Kieran Tierney has been in fine form for Celtic this season
Sir Alex Ferguson is said to have given his approval to the deal.
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The teenager is likely to command a fee of £10million.