Tierney is travelling to London this evening after a £25m deal was agreed between the two clubs.
The Scotland international is due to undergo a medical before signing on the dotted line.
Arsenal are due to pay an initial £18m plus £7m in add-ons, according to Sky Sports.
BREAKING: Arsenal have agreed a fee with Celtic over the transfer of Kieran Tierney who is on his way to London for a medical, Sky Sports News understands
— Sky Sports News (@SkySportsNews) August 7, 2019
Tierney is currently sidelined after undergoing a double hernia operation at the end of last season.
Celtic boss Neil Lennon expects the 22-year-old to return to full training next week.
When asked about Tierney’s future on Tuesday, Lennon said: “We were talking about it on the plane over here as it’s all gone a bit quiet, which is surprising for me.
“I know these deals can get done at a minute to twelve. I’d be surprised if there wasn’t another offer for Kieran, but delighted.
“He’ll 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.
“I know there is talk of him being out for another eight weeks, but that’s nonsense. I don’t know where that’s come from.
“He’s missed the whole of pre-season, but I’d say he’s more or less a week away from getting back integrated to the training.
“Add another week or ten days on top of that before he’s ready to play.”