The news comes after Gunners boss Unai Emery identified the Scotland international as his top left-back target.
The Hoops star has been a top performer in Scotland for several years and has helped his boyhood club to three domestic treble wins in a row.
The 22-year-old has been linked with a move to the Premier League for some time but a deal has never materialised.
Arsenal are in the market for defensive reinforcements this summer, with a left-back and a centre-half the top priority.
Emery knows he has to be careful with his spending this summer though after being handed a transfer budget of just £40m.
But with Tierney contracted to Celtic until 2023, Arsenal would have to table a multi-million-pound offer close to £20m for them to even consider accepting it.
According to the Daily Mail, Emery sees that outlay as a shrewd investment, with a solid bid potentially landing on Celtic’s doorstep in the coming days.
It is no surprise that there is interest in Tierney, with Leicester City and Borussia Dortmund also credited with an interest in taking him away from the Scottish Premiership.
He was restricted to just 19 league appearances last season as he struggled with a number of injuries, most notably a hip problem which kept him out for over two months from December to February.
He also had a double hernia operation at the end of May, which he had been putting off in a bid to be fit for the Scottish Cup final, which he didn’t make.
But this has not slowed down interest in the left-back though and a potential bidding war could be on the cards this summer.
Meanwhile, it is thought that transfer target Alexis Claude-Maurice favours a move to Arsenal.
It is believed that up to 10 clubs, including Newcastle, West Ham, and Lyon, are interested in the France U-20 international.
It was reported last weekend that Arsenal had made a £14m bid for the forward, who will be sold to the highest bidder.
West Ham are also believe to have made a bid last month but it was rejected.