Celtic and Arsenal have agreed a Scottish transfer record fee of £25 million for Scotland left-back
The player is set to travel south ahead of Thursday's Premier League transfer window deadline to complete the deal.
Speaking immediately after their 1-1 draw away to Cluj in Champions League qualifying, Celtic manager Neil Lennon said: "I don't know anything. I've said the last few days it wouldn't surprise me [if Arsenal came back in for the player].
"If he goes, he goes with everyone's best wishes. He's an outstanding talent and a great kid."
Tierney has won four league titles, two Scottish Cups and two League Cups for Celtic. The 22-year-old has been limited this season with a hernia injury.