The Gunners are said to have identified the Scotland international left-back as a top target for the summer transfer window.
An opening bid of around £15 million ($19m) has received short shrift, with that offer branded an “embarrassment” and “pittance” for a player of Tierney’s undoubted ability.
It has been suggested that a second approach is being readied at Emirates Stadium, with Unai Emery seeking to bring in players that will improve his side in the present and continue to develop in the future.
“He is a quality player. A team like Arsenal play football. I think it will suit Kieran and the way he plays – he is an attacking full-back. That is a strength of his.
“At the moment, they have got Monreal playing full-back. I think Kieran is a lot better than him.
“Arsenal scouts have obviously been watching him very closely over a period of time and their analysts have looked at different aspects of his play. They clearly believe he will fit in to the system that they play.
“If Kieran goes down there, stays patient and grasps his opportunity with both hands when he gets it then the sky is the limit for him.
“He isn’t stupid. He will have seen what Andy Robertson did at Liverpool. If he takes a leaf out of Andy’s book he will do well.”
Arsenal are facing up to another season without Champions League football, having finished fifth in the Premier League last season and lost in the Europa League final, but Gallacher sees signs of progress in north London.
A man who tasted title glory in England with Blackburn added: “I saw a fair bit of them last season and thought their style did change a little bit.
“They weren’t trying to walk the ball into the net as much. They were trying different strategies to attack and get that ball into the final third.
“But the players they have got are phenomenal. They have a fantastic squad, they just have to believe in each other and take it to the next level. I think you will see an improvement next season.
“Emery has been in there for a season, he understands the club, he understands the players and their qualities. All he needs is strength in depth and more consistency to get the team further up the table. He will know that and will be trying to bridge the gap.
“They have been in the top five for England for the last 25 years and are more than capable of staying there. It is a big ask for Arsenal to get the players who can help them do that. But I think Kieran Tierney would be a bargain.”