But the Gunners have the financial might to match whatever price the Scottish champions request and a move to London is thought to be on the cards.
And Celtic’s summer business has done little to quieten rumours with SK Rapid Wien left-back Boli Bolingoli closing in on a move to Parkhead, reports Sky Austria.
Bolingoli’s Rapid Wien side finished eighth last season, although they were the only team to beat runaway champions Red Bull Salzburg.
The Belgian defender is the cousin of Romelu Lukaku, who himself could be on the move to Inter Milan this summer.
Bolingoli’s potential suitors Celtic are set for a business transfer window as new manager Neil Lennon looks to embed his style.
The Northern Irishman is looking to greatly alter the club’s possession-based style that had proved hugely successful under Brendan Rodgers.
However, he had planned on building on the Celtic squad – not replacing those who leave.
Tierney was a major success story of the Rodgers-era having established himself as a leading light in Glasgow.
He spent periods of last season out with injury, but his return was highly anticipated.
But it now looks as if he could’ve played his last game in the green and white hoops with Arsenal pressing hard for his services.
Manager Unai Emery is well aware of the squad’s deficiencies, especially in defence, and is keen to address those this summer.
Tierney’s ability to play on either flank is a major plus with Emery switching between systems that require conventual full-backs, but also wing-backs.
Arsenal are also pushing to sign St.Etienne defender William Saliba, although the French club want him to return on loan with the Premier League side are opposed to.