Scottish left-back Tierney has been a target all summer for the Gunners, who have had two bids previously rejected by Celtic.
However, Tierney, who sat out Celtic's 1-1 draw with CFR Cluj in Romania on Wednesday as he recovers from a hernia operation, is set to travel to London after the clubs finally agreed on the payment structure of a £25 million move according to widespread reports.
"I've said the last few days it wouldn't surprise me," Celtic boss Neil Lennon told the BBC on the possibility of losing Tierney.
"If he goes, he goes with everyone's best wishes. He's an outstanding talent and a great kid."
Arsenal's long search for centre-back cover could also be resolved on Thursday should Luiz move across London from Chelsea.
The 32-year-old did not train with the rest of his Chelsea teammates on Wednesday.
"If we can on the last day, we will try to sign another centre-back, but one who has the capacity to help us," said Arsenal coach Unai Emery.
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