The Hoops are demanding £25m for their star left-back, who has emerged as a top target for the Gunners.
So far, Arsenal have had a £15m bid rejected and a second offer of £18m is set to fall short too.
The Gunners are reluctant to give in to Celtic’s demands as they have a limited budget to play with.
Unai Emery only has £45m plus any money from player sales this summer.
As a result, he is prioritising strengthening his defence which has clear limitations.
Tierney, 22, would serve as a long-term solution at full-back with Nacho Monreal edging nearer the end of his Arsenal career.
Manchester United are one club to have shown interest in Tierney in the past, though.
And amid Arsenal’s push to strike a deal with Celtic, the Scottish champions are doing what they can to cash in.
“It’s not a given that he will go”
Neil Lennon admitted recently the Hoops were bracing themselves for bids for Tierney.
And Celtic have shown they are ready to let their talented defender take the next step in his career by putting feelers out to other clubs over his availability.
Sky Sports say Celtic are scoping out interest from other parties who could rival Arsenal and ultimately drive the price up.
Tierney is a boyhood Celtic fan but he has already captained Scotland before his 23rd birthday and is ready for top level football in England.
Discussing the possibility of a move to Arsenal materialising, Sky Sports reporter Charles Paterson has said: “It’s not a given that he will go.
“Arsenal are a big enough club for him at this stage in his career, it would be a natural progression. The kind of football would suit him.
“It would devastate the Celtic support and it would be very hard for him – but you have to get the timing right with these moves.
“He knows at some point he’ll have to make a move if he wants to be called world-class. Does he care about that? Only he can tell you, but he’s gone on the record saying that he would be happy playing for Celtic as long as they want him.”
Serie A giants Napoli are said to be very interested in Tierney too, so Arsenal may have to act quickly to get a deal over the line.