That is according to Arsenal Fan TV host Robbie Lyle, who claims the Scotland international will complete his move to the Emirates Stadium this week.
Arsenal have had offers worth £15m and £18m rejected by Celtic, who are holding out for £25m.
Tierney returned home early from the Hoops’ pre-season training camp in Austria last week, sparking speculation he was set to join.
But the 21-year-old flew back to Glasgow to continue his rehabilitation following a double hernia operation in May.
However, Lyle claims he has heard Tierney is on the brink of finalising a switch.
“It is starting to look like Kieran Tierney will become an Arsenal player,” he said.
“The information that I’m getting is that a deal has been now agreed between Arsenal and Celtic.
“We know that Celtic have been holding out for £25m for Tierney and it does look like Arsenal have agreed now to pay that or near to that.
“The deal for Tierney should be done sometime this week.
“Although there’s been talk in other quarters that Napoli are trying to hijack the deal.
“They would like to sign Tierney and they may come in with a late.
“Tierney, apparently, wants to come to Arsenal.”
Former Scotland and Liverpool midfielder Charlie Adams believes Tierney needs to move to the Premier League for the sake of his development.
“He’s a good player, I think it’s the next progression for his career,” Adam said.
“He’s been at Celtic, he’s done amazing at Celtic in winning the treble treble.
“I think it’s probably now time for him to leave Celtic.
“I know I’m going to get a bit of stick here on Twitter but I think it’s right for him, his next step.
“And also seeing how well Andrew Roberson has progressed at Liverpool, as a player you’ve got to be thinking ‘I want some of that’.
“It’s going to be difficult for him to leave because he is a big Celtic fan.
“But if Arsenal do want him they will have to pay serious money for him.”