The Daily Mail is reporting that the north London outfit have already failed with a £15m bid for the Scotland international this summer.
The same article states that Arsenal are ready to make a bid in the region of £25m for the 22-year-old left-back in the summer transfer window.
According to the same story, the Gunners face competition from Serie A giants Napoli but the Italian club could struggle to finance a deal.
The Daily Mail go on to report that Napoli manager Carlo Ancelotti is a big admirer of the Scottish defender following his impressive performances at Parkhead.
The story adds that Ancelotti is confident that Napoli can beat Arsenal and Unai Emery to the talented Celtic youth graduate.
Tierney has won four SPL titles, two Scottish Cups and the Scottish League Cup twice since his stepping up to the first team.
Arsenal are looking to sign a left-back to bolster their defence after the Gunners conceded 53 times in 38 games last term.
The Gunners lost 4-1 to bitter rivals Chelsea FC in the Europa League final in Baku last term to miss out on a first trophy under Emery.
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