Italian giants Napoli have joined Arsenal in the race for the left-back.
Recognised around Europe as one of the best young players in his position, the Serie A runners-up have made enquiries.
According to the Scottish Sun, manager Carlo Ancelotti is a big fan of the 22-year-old.
Tierney, who is contracted until 2023, has already made 170 appearances for Celtic, 42 of which have come in European competition where he has faced and the likes of Lionel Messi, Arjen Robben and Raheem Sterling.
Napoli already have two international left-backs on their book in Faouzi Ghoulam and Mario Rui but will compete with Arsenal for the £26-million rate defender.
Gunners boss Unai Emery sees left-back as a priority and Tierney his No.1 target.
The Premier League side may use fringe players as a makeweight in the deal.