The Hoops value their defender at around £25million and the Gunners met that price in their most recent bid.
However, Celtic were not happy with the structure of the deal, particularly an incentive that was based on Arsenal reaching the Champions League.
It is understood the Scottish club want the vast majority of the fee paid up front, and according to the Daily Record, the Gunners will launch a third attempt to satisfy these demands.
Tierney has four years left on his deal at Celtic Park and the Hoops are under no pressure to sell.
Manager Neil Lennon said last week: “He has a valuation and the way the market is in England it could even look like a bargain when you think of [Aaron] Wan-Bissaka going for £50m.
“We feel Kieran is a far more experienced and rounded full-back at this juncture in his career.
“We’re certainly not going to be pushed over in any negotiations.
“There’s no question Kieran could play in the Premier League. We’ve had [Virgil] van Dijk, [Victor] Wanyama, [Fraser] Forster, [Gary] Hooper, [Moussa] Dembele – all these players – and he is as good.
“His rise has been meteoric. He is a top quality player, one of the best as in Britain. The transfer fee should merit that.
“We are at the 20-plus and it’s been rejected out of hand. It’s up to Arsenal or anyone else to reach our valuation of Kieran.”