The Gunners first registered an interest in the left-back last month when they submitted a bid of around £15million.
However the offer was rejected by the Scottish champions who believe the offer was well short of their valuation.
Now Arsenal have submitted a new offer of £25m and are waiting to hear back from Celtic.
The player is also attracting interest from Napoli.
They said: “Kieran loves Celtic but his heart is set on a move to the Premier League.
“We all know he wants to further his career and offers like this do not come along often, so it looks inevitable.”
The fee would be a Scottish record, surpassing the £19.7m Lyon paid them for Moussa Dembele.
Neil Lennon already has a replacement in place in the shape of Boli Bolingoli-Mbombo, who recently joined the club from Rapid Vienna.
The Northern Irishman has revealed that Tierney is currently suffering with a pelvic injury but it is not expected to impact the deal.
“He’s still injured. He is rehabbing,” he said.
“He had the double hernia and he still has the osteitis pubis problem, so it’s a slow burner.”
Unai Emery is keen to bring the flying Scotsman to the Emirates Stadium after losing Nacho Monreal on the expiry of his contract.
The Spaniard is working on a limited budget this summer and is expected to ask Stan Kroenke for more funds.
They were on the verge of sealing the signing of William Saliba from St Etienne before Tottenham attempted to hijack the deal.
Crystal Palace appear to have priced Arsenal out of a move for Wilfried Zaha as they demand in excess of £70m.