Unai Emery has identified Tierney, 22, as his No.1 left-back target and the Scotland defender is keen on the move.
The Gunners have had an opening £15m bid knocked back but are hopeful of reaching a compromise with Celtic.
Arsenal are set to return with an improved £19m offer for Celtic star Kieran Tierney
Tierney saw a move to Everton fall through last summer and has interest from Napoli too this time round.
The 22-year-old Tierney has had hip and hernia issues in recent months and subsequently underwent surgery for a double hernia at the end of May. He is hopeful, however, of being fit for pre-season.
Emery knows he must improve his Arsenal squad this summer on a shoestring budget, with Sportsmail revealing the Spaniard will be permitted £45m plus additional funds raised by player sales.
Arsenal boss Unai Emery has made Tierney his No 1 priority signing at left-back
Missing out on the Champions League has hindered the club, who have not benefited from the financial windfall provided by European football’s premier competition.
Emery is on the hunt for defenders following the absence of Hector Bellerin, who will miss the start of the coming campaign due to repeat issues with a knee injury.
On the left side of defence, Emery spent last season relying on veteran full-back Nacho Monreal and Sead Kolasinac.