BBC Sport's Brian McLauchlin reported an agreement has been reached after the Hoops previously turned down two bids from the Gunners. He also noted how Tierney will now travel from Glasgow to London to "discuss personal terms."
Arsenal have been on the case of Tierney throughout this transfer window. The Gunners saw an offer of £15 million knocked back, before returning with a bid of £25 million in July, only for the Hoops to bemoan the number of add-ons in the proposed deal.
Celtic wanted more money up front, and it seems Arsenal have met that request. It makes sense to stump up the cash for a dynamic young player who can be an asset for the Gunners at both ends of the pitch.
Still just 22, Tierney is an impressive defender who also offers pace going forward and technique in attacking areas. His final ball would be a major upgrade over the service provided by Sead Kolasinac.
Overlapping full-backs are key to the enterprising high-press Unai Emery has tried to install at Arsenal. The approach not only demands energy going forward, it also requires the athleticism to track back quickly and cover gaps in defence, quickness 33-year-old Nacho Monreal doesn't possess.
Tierney can also operate at centre-back, a position that remains a glaring weakness in Emery's squad. It's a problem that's taken Arsenal to considering a move for Chelsea's David Luiz before the deadline on Thursday, per David Ornstein and Simon Stone of BBC Sport.
Landing Tierney would bring to an end a protracted and thorough pursuit of a target the Gunners have long admired:
Passing a medical remains the next step, a potentially daunting one since Tierney has endured his share of injury problems. Celtic boss Neil Lennon recently said Tierney will "be fit in a week or two. He still has some abductor pain, but I think the pubic problem is starting to settle and he's over the double hernia," per Craig Swan of the Daily Record.
If all goes well, Arsenal will get a player who can bolster the defence in a variety of ways, but Celtic appear sure to miss Tierney. The Hoops were given an uncomfortable glimpse of what life without him will be like during Wednesday's 1-1 draw away to Cluj in the UEFA Champions League qualifying phase:
Securing a deal at long last would cap a solid window for the Gunners. The north London club have already broken their transfer record to recruit prolific winger Nicolas Pepe and acquired Real Madrid playmaker Dani Ceballos on loan.
Tierney could become the next young cornerstone of a defence set to be underpinned by imposing centre-back William Saliba in 2020, after Arsenal signed the Frenchman and loaned him back to Saint-Etienne for this season.