The left-back is on his way to London and looks set to sign ahead of Thursday's Premier League deadline.
Arsenal's initial £15m bid for Tierney was rejected in June, as was a second £25m offer, as it contained too many add-ons and was dependant on the Gunners qualifying for the Champions League.
The 22-year-old, who has won five Scottish Premiership titles, made 40 appearances in all competitions for the Hoops during the 2018/19 campaign.
If Arsenal are able to complete the deal for Tierney he would be Unai Emery's fifth signing of the summer window.
