The Gunners have been relatively quiet in the summer transfer window thus far and Unai Emery has been held been held back by the restrictive budget which has been put in place following their failure to qualify for the Champions League last season.
Despite their lack of funds, Arsenal have been pursuing a number of targets, including Tierney, who was the subject of a £25m bid last week - a bid that was swiftly turned down by Celtic.
Celtic were looking to secure the bulk of that fee up front and, according to a report from the Daily Record, Arsenal are now willing to open up negotiations for a third time as they remain unflinching in their interest.
Emery is said to be confident that Arsenal can secure a deal for Tierney before the transfer window closes on August 8, which could indicate that Arsenal will be willing to match Celtics terms in order to secure a deal as soon as possible.
Arsenal have a number of other avenues to explore over the coming weeks and, as revealed by Emery, they are still looking to improve their squad as much as possible.
"Some players are very expensive and we are waiting for this possibility," he recently told Sky Sports.
"We dont want now to go to the second line of players. We are waiting. I want to sign one, two, three or four players only if we can be sure they are coming to improve our team."
Arsenal also remain in the hunt for the likes of Wilfried Zaha and William Saliba, but there have been no other updates regarding their pursuit of the pair.