The Gunners have made Tierney a priority signing this summer.
The Celtic defender is one of the most highly-rated talents north of the border, having won the three consecutive SPFA Young Player of the Year awards prior to last season.
Tierney is currently undergoing rehabilitation from a double hernia operation he had in May, having only managed to play a handful of matches since the start of December.
That is not thought to have put Arsenal off though, with the Gunners already launching a £15m bid.
That offer was turned down, however, with Hoops boss Neil Lennon stating their valuation was well off the mark.
Emirates bosses are expected to go back in for Tierney in the coming days with an improved offer of £19m, still short of Celtic’s reported asking price of £25m.
However, according to the Metro, they will face competition from Napoli.
They claim the Serie A side watched Tierney extensively last season, and boss Carlo Ancelotti has been highly impressed from what he has heard.
Napoli will have to act fast though, and are understood to be weighing up an offer they feel will hijack the Gunners’ approach.
Meanwhile, Celtic and Arsenal legend Charlie Nicholas thinks a move to London would be good for Tierney if he wants to play in a “bigger league”.
“I thought Kieran Tierney would want to stay at Celtic,” he told Sky Sports.
“He’s a die-hard Celtic fan, they’ve got eight in a row wrapped up, they want to go and do nine, and I thought if Kieran really wanted it, if he does nine then he’ll want to do 10 and get in the history books.
“But the stories I’m hearing here in Glasgow is that he might be keen to move on to the bigger league. That’s really what it’s about, a bigger league.
“Will it make him a better player? Probably. There is a question mark about his defensive frailties.
“But then again, he would be an improvement for Arsenal – there’s no doubt in my mind about that.”