The Daily Telegraph is reporting that the Scottish champions have already rejected two bids from the Gunners for the 22-year-old left-back this summer.
Both of those bids are believed to be less than £19m, as Unai Emery sets about trying to bolster his squad ahead of next season.
According to the same story, Celtic value Tierney at closer to £25m, which would make him the most expensive player ever sold by the SPL club.
The same article says that Arsenal only have a transfer budget of around £45m available to them this summer as they look to strengthen their squad.
However, the story also confirms that Arsenal will be able to re-use any funds generated from player sales in the current transfer window.
The same article says that Arsenal would be able to offer Tierney wages of around £70,000 a week.
Arsenal are believed to be on the lookout for a long-term replacement for Nacho Monreal, who turned 33 earlier this year.
The Gunners finished in fifth place in the Premier League table and without a trophy last term.
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