The Gunners have had an initial bid of £15million thrown out by the Scottish champions with the fee falling far below the club's valuation.
The Premier League side are expected to return with a bigger offer, most likely a package worth more than £20million.
With a reduced budget after missing out on Champions League football, Unai Emery is expected to spend the vast majority on the Scotland international and St Etienne centre-back William Saliba.
Tierney is currently in Austria preparing for the club's Champions League first round qualifier against FK Sarajevo.
Manager Neil Lennon admitted that the player, as all do, has a price but warned Arsenal to be serious with their next offer with the club keen on holding onto him.
"Arsenal are aware of what we value the player at.
"He’s on a long-term contract, he’s an asset and we don’t want to sell him.
"It’s going to have to take a lot more than what they’ve offered the first time around to give us a decision to make."