But Unai Emery is keen to secure the services of the Scotland global with a new left-back one of his top priorities.
The 22-year-old, who already has 170 appearances for Celtic under his belt, is now recovering from hernia surgery, although it is believed that he could recover in time for the beginning of the upcoming season.
"We all know he wants to further his career and offers like this do not come along often, so it looks inevitable".
Tierney is still recovering from double hernia operation but this is unlikely to prevent a move from happening.
Celtic started their season in fine style in Sarajevo on Tuesday evening, without Tierney, and they will be excited with what they can continue to achieve when their defender is back from injury.
"He had the double hernia and he still has the osteitis pubis problem, so it's a slow burner".
The question now is how much are Celtic willing to push up their asking price for the defender without scaring away Arsenal?
The Scottish champions don't appear to have responded to Arsenal's latest offer but it will be hard for them to turn down £25m so Tierney could be about to become the most expensive player in Scottish history.
The fee would be a Scottish record, surpassing the £19.7m Lyon paid them for Moussa Dembele.
Unai Emery's men are wrestling with a tight transfer budget this summer.
Kieran Tierney of Celtic with the League Trophy during the Ladbrokes Scottish Premiership match between Celtic FC and Heart of Midlothian FC at Celtic Park on May 19, 2019 in Glasgow, Scotland.
Tierney has been at the centre of transfer speculation since the window opened in June, with Arsenal having a £15m bid rejected and Napoli known to have an interest in him too.