Arsenal boss Unai Emery has made Tierney one of his main targets this summer.
The Gunners latest bid for the 22-year-old could reach up to £25m, but Celtic have rebuffed the offer.
The current state of deadlock between the two parties is down to the structure of the deal, reports the Mirror.
Arsenal’s transfer kitty is relatively minimal in comparison to their rivals meaning they have to be creative with their offers throughout the window.
The Gunners want to pay in instalments and include add-ons for a player who has had his fair share of injuries.
Celtic, meanwhile, want the whole sum upfront which Arsenal are not willing, or may not be able, to do.
They are keen to maximise the value of one of their prized assets having been shortchanged in recent years.
The Scottish champions first put Virgil van Dijk on the map when they signed him in 2013.
He moved on to Southampton three years later for £13m, before becoming the world’s most expensive defender 18 months ago.
They are therefore looking to avoid making a similar mistake and know full well Arsenal are in the market for a full-back.
The Gunners, meanwhile, are closing in on a £27m deal for Saint-Etienne defender William Saliba.
An agreement is in place which would see the Frenchman remain at the Ligue 1 club for another year on loan before heading to north London.
Emery is also looking to add a winger this summer and has looked to move for wantaway Crystal Palace star Wilfried Zaha.
The Ivorian has admitted he wants to leave Selhurst Park and is a boyhood Arsenal fan.
But the Gunners £40m bid is only half way towards Palace’s £80m valuation of the player.