According to Sky Sports, the Scottish international is on his way to London for a medical and the deal is believed to be worth around £25 million, per BBC Sport.
It should be completed on Deadline Day.
Tierney will join forward Gabriel Martinelli, winger Nicolas Pepe, centre-back William Saliba and midfielder Dani Ceballos as Unai Emery's summer arrivals.
After losing captain Laurent Koscielny to Bordeaux earlier this week, his arrival will be a major boost to Arsenal's backline.
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The move will also put an end to one of the summer's longest-running transfer sagas.
The 22-year-old has been courted by the Gunners since the conclusion of the 2018/19 season.
Celtic are reported to have rejected two bids over the last few months but have finally accepted a Scottish-record £25 million.
There have been some fears over Tierney's fitness.
Although he impressed throughout the last campaign, the youngster suffered a hip injury that meant that he was ruled out of action for two months between December and February.
Just last week, Neil Lennon revealed that he's also been recovering from a double hernia.
"He's still injured and he's rehabbing," the Hoops boss said on August 3.
"He had a double hernia and he still has the Osteitis pubis problem so it's a slow burner that one."
Despite carrying an injury that could see him ruled out for a few more weeks, Arsenal have pushed on and will attempt to secure his signature by Thursday's deadline.
As well as completing a deal for Tierney, the north London side have also been linked with a sensational move for Chelsea centre-back David Luiz.
Reports suggested on Wednesday that the Brazilian has refused to train with the Blues in an attempt to force through his move.
With both Luiz and Tierney rumoured to be pushing for transfers, it could be a very exciting Deadline Day for Arsenal fans.News Now - Sport News