The Gunners’ opening bid of £15m was dismissed by the Scottish champions, who also knocked back a second offer of £18m.
Arsenal then increased their bid to around £25m in recent days, but that too was turned down by Celtic.
The Bhoys were unimpressed by the terms of the deal, which featured numerous add-on payments and relatively little money up front.
Despite their three rejections, Arsenal are still hopeful of landing the left-back this summer.
However, Unai Emery may first need to trim his squad to raise funds for any new signings, since Arsenal’s current transfer budget is just £40m.
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