Arsenal stepped up their interest in Tierney, 22, over the weekend and are understood to have submitted an opening bid of £15million.
Express Sport understands Celtic are looking for closer to £25m but the Gunners are willing to offer Carl Jenkinson in exchange to encourage them to soften their stance.
Now, The Sun claim the Hoops have already started preparing for his departure by turning their attention to Iran international Milad Mohammadi to replace the Scottish star.
Celtic could now snap up Mohammadi, who is out of contract after turning down a new deal to stay at Russian outfit Akhmat Grozny.
Fans might remember Mohammadi from last summer’s World Cup where he shot to fame with his failed attempt at a forward roll while taking a throw-in.
Meanwhile, Tierney is said to be keen on the move and reports say he has asked for permission to talk to interested clubs.
He was linked with Everton last season although they no longer have a need for a left-back thanks to Lucas Digne’s form since signing from Barcelona.
Arsenal, though, are desperate for reinforcements after a disappointing end to last season.
Their failure to qualify for the Champions League means boss Unai Emery is working with a tight budget of around £40m so the Spaniard will be keen to get a deal done as cheaply as possible.
He needs fresh blood all over the pitch but defence remains his highest priority area.
The absence of a top-class centre-back became was evident for most of last season and compounded during their run-in at the end of 2018-19 that included a humiliating 4-1 defeat by Chelsea in the Europa League final.
Emery also wants a new right-back because of Hector Bellerin’s injury problems and the departure of Stephan Lichtsteiner.
He is also yet to sign replacement for Aaron Ramsey in midfield after his free transfer to Juventus.
Arsenal are well stocked in attack but lack cover in wide areas and Yannick Carrasco is a long-term target.
While Emery has limited funds to spend, he can increase his budget with player sales.
Mohamed Elneny is set to leave the club and a good showing for Egypt in the Africa Cup of Nations could attract some interest.
Lucas Torrerira is another who is unsettled after he admitted he was struggling to adapt to life in England.