The Gunners have been chasing Tierney all summer but have so far fallen short with their bids for the full-back.
Napoli are stepping up their attempts to land sign the Scot by preparing to sell current full-back Mario Rui.
Carlo Ancelotti has told the player he is surplus to requirements, according to the Telegraph.
Rui, 28, will bring in around £15m – and the Italian side plan to use the funds to sign Tierney in the coming weeks.
Arsenal are still confident of landing the 22-year-old, despite Unai Emery’s limited budget this summer.
The Spaniard has around £45m to spend, plus whatever revenue is brought in from player sales.
Arsenal could have done without Napoli announcing their interest as Celtic could now hold out for a bidding war, which the Serie A side will probably win.
The Scottish champions are said to be happy for Tierney to leave, should an acceptable bid be launched.
Allan Preston, who represents the Scotland captain, insists Celtic hold all the cards at present.
“It’s all down to Celtic,” Preston told the BBC.
“If they get a price that they believe is the correct value he could possibly be on his way.
“No one has hit that price yet.
“Will there be a time for Kieran Tierney to go? I think there will be.
“If it’s just now, it will be up to the clubs involved, Arsenal, Napoli or whoever to then put the money on the table and make Celtic think about this.
“The ball is firmly in Celtic’s court at the moment.”
Arsenal are also pursuing a move for William Saliba.
The Saint-Etienne defender will cost around £20m before being loaned back to the Ligue 1 side.