The Gunners have been linked with a move to sign the Scotland international over the past couple of weeks but a deal has failed to materialise so far.
Arsenal were reported to have failed with an opening bid of £15m for the 22-year-old full-back as Unai Emery looks to improve a key area of his team.
The north London side are understood to have limited funds to spend in the summer transfer window to improve their squad ahead of the new Premier League season.
Sky Sports News reporter Solhekol has now taken to Twitter to report that Arsenal have submitted a second offer for Tierney in the summer transfer window.
Solhekol wrote on Twitter: “Arsenal have made £25m bid for Celtic’s Kieran Tierney. If their transfer budget this summer is really £45m, that would leave £20m to buy Wilfried Zaha.”
Arsenal have only signed Brazilian teenager Gabrielle Martinelli in a £1.6m deal from South American club Ituano in the summer transfer window so far.
The Gunners finished in fifth place in the Premier League table last season to miss out on a spot in the Champions League for the third successive campaign.
Arsenal appointed Unai Emery as their new manager last summer after Arsene Wenger left the Emirates Stadium outfit following 22 years in charge of the north London club.
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