The Gunners failed in their initial attempt to persuade the Scottish Premiership champions sell Tierney for £15m.
They have boosted their offer but are understood to still be some way short of the Hoops’ £25m valuation.
Manchester United are currently negotiating with Crystal Palace for full-back Aaron Wan-Bissaka.
And with the Eagles’ star’s price nearing £50m, The Times reports that Celtic are prepared to play hardball with Arsenal over Tierney.
The 22-year-old has captained the Bhoys amongst his 167 appearances and has also represented Scotland at senior level.
Former Celtic goalkeeper Pat Bonner has questioned whether Arsenal is “a big enough club” for Tierney.
He believes a fee of “around £25m, with add-ons” would be “a price you’d start talking about”.
“If Kieran was playing in England, he’d probably be worth about £50m,” he said.
“But because it’s Scotland, I would think £25m would be a price you’d start talking about, with add-ons.
“He’s got the ability to go into a club like Arsenal. Whether Arsenal is a big enough club for Kieran Tierney, that’s the question.
“If you were picking out a player in British football at the moment that can play left-back, himself and [Liverpool’s Scotland captain] Andy Robertson are the best two in Britain, if not Europe.”
Former Everton and Scotland striker James McFadden says the £15m figure is not “anywhere near enough” for a player of Tierney’s calibre.
“I don’t think that will be even close to tempting Celtic into selling,” he added.
“You only need to look at his rival in the Scotland side, Andy Robertson, to say that there’s not much to pick between them and for me they are as good as each other.
“So that would tell you Celtic would be looking for far more if they want to sell him.
“I imagine at some point it would be good to see him test himself somewhere else but I don’t think that point is right now.”