The Gunners have been heavily linked with a move to bring the 22-year-old left-back to The Emirates this summer as Unai Emery looks to strengthen his squad.
However, Arsenal have so far been unsuccessful in their attempts to bring the Scot to north London, with the Gunners having struggled to agree a fee with Celtic.
Now, BBC Sport reporter McLaughlin has confirmed that Arsenal have had a fresh bid worth up to £25m rejected by the Scottish club.
Writing on Twitter, McLaughlin said: “#Celtic knock back #Arsenal’s latest bid for Kieran Tierney. Second offer was worth up to £25 million.”
Tierney made 21 SPL appearances for Celtic last season and also featured five times in domestic cup competitions.
Arsenal have so far only made one signing this summer after having wrapped up a deal for Brazilian teenager Gabriel Martinelli.
The north London side are believed to be on the lookout for a number of new signings as they prepare for Emery’s second campaign in charge.
Arsenal finished in fifth place in the Premier League table last term and without a trophy, and they also lost the Europa League final 4-1 to Chelsea FC as they missed out on Champions League qualification.
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