The Gunners are in need of several reinforcements and Unai Emery has lodged a £15m bid for the 22-year-old.
But former Parkhead star Chris Sutton believes the Scottish champions cannot afford to make the same mistake they did with van Dijk.
Young English star Aaron Wan-Bissaka is set for a £50m move to Manchester United while Leicester are demanding £70m for Ben Chilwell.
And Sutton believes the comparative quality between Tierney and his two English alternatives means Celtic should be demanding fair money.
“Manchester United are paying £55m to land Aaron Wan-Bissaka from Crystal Palace,” he wrote in the Daily Record.
“Leicester are looking to get about £70m for Ben Chilwell if he goes.
“Both of those boys have had roughly one good season each in the English Premier League and that’s the fees?
“Are you trying to tell me that Chilwell is a four times better player than Tierney? Or that Wan-Bissaka is three times as good?”
Tierney broke into the Celtic team under Brendan Rodgers and has since gone from strength to strength.
The Bhoys have previously had their stars cherrypicked by teams in England with van Dijk and Victor Wanyama two major examples.
But Sutton doesn’t think Celtic should be accepting less than the market value for their star players, insisting the £13m they accepted for Van Dijk from Southampton was hugely detrimental to the club.
The Dutchman’s success since moving in 2015 serves to demonstrate how Celtic could have demanded far more money.
He added: “These are exactly the same discussions which took place when van Dijk was being touted for £13m at the same time as John Stones was leaving Everton to join Manchester City for the guts of £50m.
“I made it quite clear back then it was daylight robbery. Stones was never four times the player that Van Dijk was.
“That’s not hindsight. I said it on these very pages at the time.
“Celtic undersold themselves badly in the van Dijk deal. That’s been shown up subsequently.”