The Italian club had reportedly looked at the Scotland defender but they do not intend to make a bid, leaving the path clear for Arsenal to take him to England if they can agree a deal with Celtic.
Napoli manager Carlo Ancelotti, speaking after his side’s 3-0 victory over Liverpool at Murrayfield, said: “I know him and he is a really good player, the Celtic left-back.
“We are looking for a lot of players during the season. Just now he is a Celtic player. We have two good left-backs but he is a really good player.”
Asked if he was therefore interested, Ancelotti replied: “No.”
Arsenal have already had two bids for Tierney rejected but are confident they can strike a deal.