The Hoops star is a top target for the Gunners and around half of their summer transfer budget could be spent on him.
Tierney would slot straight in at left-back in Unai Emery’s team.
He would be a long-term solution in the role with Nacho Monreal, 33, in the latter stage of his Emirates Stadium career.
Sead Kolasinac, meanwhile, has not set the world alight since arriving at Arsenal two years ago.
But while the signing of Tierney makes sense on paper, Arsenal’s limited transfer budget leaves them stretched.
With only £40m left to spend on strengthening the squad, before any player sales, Emery has to choose carefully.
A winger and a centre-half are greatly desired too, with a £25m deal for Saint-Etienne’s William Saliba on the verge of being sealed.
The young central defender is set to remain in France until the start of the 2020/21 as part of that deal, though.
“A deal can be done swiftly as and when he is needed”
With regards to Tierney, Arsenal are close too, with Football London suggesting a deal can be done swiftly as and when he is needed.
The huge £80m price-tag surrounding Wilfried Zaha and need for further additions has left Arsenal in limbo over whether to up their offer for Tierney.
A £15m bid was initially rejected before an £18m approach failed to, with Celtic valuing the Scotland international at £25m.
Napoli’s interest in Italy has complicated matters for the Gunners as well.
They face the prospect of losing out to the Serie A giants if they do not move quickly.
However, Football London add Tierney is believed to favour a move to north London over Naples.
Arsenal are expected to table 22-year-old Tierney a contract worth £70,000-a-week.
And a compromise could yet be reached relatively quickly with Celtic for a fee just in excess of £20m.
When Arsenal will ultimately decide on finalising a swoop for Tierney, though, remains to be seen.
Attention over the last few days appears to have centred around a winger.
Ivory Coast international Zaha, who is still away at the Africa Cup of Nations, is still proving too expensive even with Arsenal willing to offer a player-plus-cash deal to Palace.
As a result, Starsport understands interest in signing £20m Ryan Fraser is now being explored again, with Bournemouth potentially cashing in on the Scot who has refused to sign a new contract.