Arsenal agreed a fee with Celtic on Wednesday for the Scotland international left-back.
Arsenal’s initial £15m bid for Tierney was rejected in June as was the second 25m offer due to too many add-ons and was dependent 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.
Tierney would be Unai Emery’s fifth signing of the summer window at the end of a successful medical.