According to the Daily Mail's Simon Jones, the Gunners have restructured the deal, with Celtic willing to sell the left-back, provided they receive more of the fee up front.
Tierney is said to have told his club he wants the opportunity to play in the Premier League.
The move had appeared to be in jeopardy last week, with the north London outfit's offer £5 million below the Scottish champions' valuation, and Arsenal were also unwilling to increase the proportion of the fee that would be offered as an initial payment.
Football.London's James Benge believes Tierney would be an excellent addition to Arsenal's squad:
Nacho Monreal turned 33 in February, while Sead Kolasinac is yet to convince defensively, so a promising new left-back would address a problem area in the squad.
Squawka also feels he would be a shrewd purchase from the Gunners:
Tierney only turned 22 in June, but he has been a key fixture in Celtic's back line since the 2015-16 season.
In that time, he has helped the Scottish giants win four consecutive Premiership titles, with each of the last three coming as part of a domestic treble.
With eight goals and 37 assists to his name for Celtic in 170 appearances, it's clear he has much to contribute going forward.
He's also made 12 appearances for the Scotland national team, where he's unfortunate to find that perhaps the only player better than him in the side is fellow left-back and Liverpool star Andy Robertson.
Tierney wouldn't have that issue at Arsenal, though, so he'd likely be the first choice.
He seems ready to make the step up from Scottish football, too, and at a side like Arsenal he could reach new heights in the coming years.