The Gunners saw a £15 million ($19m) offer knocked back for the Scotland international in June and had been widely expected to return with an improved bid to try and tempt the Scottish champions into doing business.
And they have now made their move, with an offer that totals around £25m ($31m), although it is once again structured heavily with installments and add-ons to get it up to the sort of figure that Celtic have been demanding.
Arsenal are hoping the offer is attractive enough for Celtic, who are yet to respond to the bid.
Tierney is currently recovering from a double hernia and is also battling a pelvic issue, with Hoops boss Neil Lennon revealing last week that the defender was still a month away from full fitness.
Speaking about the possibility of the 22-year-old moving to Arsenal, Lennon added: "I have taken no notice of the headlines. I am well aware of where we are in terms of the situation.
"I spoke to Kieran in Austria and to be honest I haven’t really seen much of him because he came back early for rehab, so has not been on the training ground as yet.
"It can be an unsettling period for any player and that is understandable but until a club comes up with any sort of realistic valuation of the player we don’t have to do anything and we won’t.
"It’s as you were. Kieran is getting on with his rehabilitation and we are getting on with what we have to do here."
Arsenal are currently in the United States preparing for the first of four games during their pre-season tour.
The club’s new technical director, Edu, is with the squad in Los Angeles - as is head of football Raul Sanllehi.
Arsenal had agreed a £27m ($34m) deal for the 18-year-old , who had agreed a five-year contract.
But just as the transfer was being finalised, Spurs matched Arsenal’s overall offer, while also offering a more attractive initial package to Saint-Etienne in terms of how the deal would be structured.
Talks have continued over the weekend, but Arsenal remain hopeful they have already done enough to convince the player to a move to the Emirates.