The Gunners have been heavily linked with a move to sign the Scotland international throughout the 2019 summer transfer window.
Arsenal are reported to have made three bids for the 22-year-old but the Scottish Premier League champions want £30m up front for the full-back.
The north London side are working with a small summer transfer budget, which has made it difficult for Unai Emery to get deals over the line.
Tierney’s asking price would blow through most of the Spanish head coach’s funds to bolster his Arsenal squad ahead of the new Premier League season.
However, Daily Record reporter Castles believes Arsenal are edging closer to a potential deal to sign the promising Scottish left-back.
“As we’ve said, Celtic will sell if the money is right,” Castles told The Transfer Window podcast.
“They do work on the premise that they need to bring money in from transfers each summer to keep their budget running over.
“I believe they’ve increased their wage bill quite significantly in recent years, apparently as a result of having Brendan Rodgers as manager. Therefore, there’s an incentive on their side to sell.
“It seems Arenal are getting closer to the price they’re looking for.
“They’ve still got a bit of time in this window in that they’ve got till the 8th of August to complete that deal, so you would think that they can get this one done.”
Tierney has won four Scottish Premier League titles as well as the Scottish Cup and Scottish League Cup twice in his fledgling career so far.
The Scottish defender has netted eight times in 170 games throughout his Celtic career, including 41 appearances in Uefa competitions so far.
Tierney broke into the Celtic team under Brendan Rodgers but the Hoops have managed to keep their prized asset at Parkhead.
Arsenal finished in fifth place in the Premier League last term to condemn the Gunners to another season in the Europa League.
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