It was the Gunners manager who personally urged club chiefs to revise and significantly improve their opening offer.
Arsenal had a £15m bid rejected by Celtic earlier in the window, and they have now returned with a £25m proposal.
While that figure does match the asking price set by the Scottish champions, it has been turned down.
This is down to the payment structure set out in the bid which Celtic did not approve of.
The latest offer included a series of staggered clauses and trigger points.
While they could eventually pump up Tierney’s price over time, Celtic are intent on getting more money up front for their talented left-back.
They do not want to be depending on the 22-year-old having success down the line at the Emirates Stadium.
As a result, Arsenal will have to make an improved offer once again.
But despite the latest setback, the Scottish Daily Star understands a deal is inching towards completion.
Serie A giants Napoli are continuing to monitor Tierney’s situation, providing Arsenal will competition for his signature.
However, it is said Emery is getting closer to landing the player after demanding Arsenal step up their hunt.
Scotland international Tierney would provide the Spaniard with a long-term solution at left-back as well as offering versatility.
Tierney has shown he can play as a wing-back and on the left of a three-man defence.
Arsenal must get a deal over the line first, though, and fans have been left frustrated with the slow progress the club has made in the transfer window.
Off the back of another failed bid to secure Champions League football, Arsenal are yet to make any senior signings to strengthen their squad, having only landed 18-year-old forward Gabriel Martinelli for £5m from Ituano so far.
“They do not have a single acceptable offer on the table”
A centre-half and a winger are still wanted on top of Tierney, who would take up more than half of Arsenal’s remaining budget for the summer at £25m.
The Gunners will need to pay more of that figure up front if they are to convince Celtic to sell.
Discussing the current state of affairs, a source told The Scottish Daily Star: “The best way of putting it is that, even if Celtic wanted to sell Kieran – which they don’t – they do not have a single acceptable offer on the table.”
It remains to be seen when Arsenal will table a third bid with the squad now in the US on their pre-season tour.