The Gunners want the left-back as Unai Emery looks to revitalise the fallen giants.
The Londoners haven't participated in the Champions League since 2017 having seen Premier League rivals jump ahead of them.
And it is this inability to qualify for Europe's premier competition that has become a sticking point with the transfer offer.
After having an initial £15million bid for Tierney rejected, Arsenal increased their package to £25million made up of a number of adds-on, including the club qualifying for the Champions League.
According to Sky Sports, Celtic have their doubts of the Gunners' ability to reach the top four in the Premier League to ensure qualification.
Tierney, who is still building up his fitness after his double hernia surgery, remains a key target with reports suggesting that Arsenal are confident a deal can be done for the Scotland international.
The Daily Record report a third offer will arrive this week.