The North London club have had two bids turned down for the player, the first of which was believed to be in the £15 million range.
While the second bid was said to have matched Celtic's demand of £25 million in total valuation, it came with a number of clauses and additional fees that didn't satisfy Celtic.
Mills reckons between £30 million and £40 million would be a fair price for the Scottish international and that Arsenal aren't showing themselves to be particularly determined about signing Tierney.
He told talkSPORT: "Realistically £35-40m for Tierney would be about the ball park.
"When you go in ridiculously low - if you want a player but go in low it doesn’t say you really want this player.
"Wan Bissaka was ridiculously over-priced. I’d say £35m-£40m for Tierney would be fair."