The left-back has been heavily linked with a move to the Emirates in recent weeks.
However, Napoli are also chasing Tierney this summer.
The Scotland international is wanted as a like-for-like replacement for Nacho Monreal in this window.
And according to a source close to the player, a move to Arsenal is looking likely.
CalcioNapoli24 report that an insider has revealed Tierney is more likely to join the Gunners than move to Serie A this summer.
But the same source also confirmed Napoli’s interest was genuine and talks have been held.
Arsenal are desperate for reinforcements this summer after finishing fifth last season.
But Unai Emery has a small budget to work with in this window, meaning he is having to pick and choose between top targets.
Arsenal have also been linked with a move for Crystal Palace’s Wilfried Zaha.
But the Ivorian will cost no less than £80m this summer, which would obliterate the north London club’s budget.
And Arsenal legend David Seaman thinks the club should prioritise signing Tierney in the near future.
“Yes, Zaha is a good player who could do well for the club, and yes I know I would say as a former goalkeeper that the defence is the first place to look at in terms of signing new players – but the simple fact is that the backline needs strengthening,” Seaman told Football London.
“I think Tierney would be an ideal addition for Arsenal.
“He’s young, he’s talented.
“I think I’m right in saying he’s already captained his club and country and by all accounts he’s a hard worker with a great attitude.
“Being British he’d also settle in straight away and there would be no period of adjustment.
“I really think they should be looking to sign Tierney before setting their sights on Zaha.”