The 32-year-old Brazil global, who signed a new two-year Chelsea deal in May, will have a medical at Arsenal on Thursday.
The outlet adds that Tierney and his agent Allan Preston were earlier spotted at Edinburgh Airport awaiting a flight to London.
The Gunners on Wednesday, agreed a £25million fee with Celtic for the left-back.
David Luiz has completed a shock move from Chelsea to Arsenal in an £8 million ($10m) deal.
Tierney, who has captained both the Hoops and Scotland, leaves behind his boyhood club to join the Gunners.
The Gunners boss acknowledged this, saying: "David has huge experience and I look forward to working with him again". I'm delighted to be here.
Arsenal will be without Mesut Ozil and Sead Kolasinac for their opening game of the Premier League season following "further security incidents", the club has said. Unai Emery was very keen to bring in the Scotsman to the Emirates for the right cost and eventually they have succeeded in clinching the deal at the third attempt.
Arsenal fans will have to wait to get their first glimpse of him in action as the Scottish left-back is now injured. Progressively I want them to feel good with their partners, with their coaches, with us.