Celtic have already knocked back an opening offer worth £15m from Arsenal for the 22-year-old left-back.
The Scottish Premiership champions are looking for substantially more – they value Tierney at around £25m.
Arsenal, however, believe a compromise can be reached between the two clubs.
Gunners bosses are reportedly preparing a new offer worth £18m.
Celtic are likely to demand more because of Tierney’s status as one of their brightest talents.
But, according to the Telegraph, Arsenal are optimistic a transfer fee will eventually be agreed.
Unai Emery is eager to add Tierney to his squad for his attacking endeavour from full-back.
The Gunners boss reportedly believes the Scotland international can make a similar impact to Andrew Robertson at Liverpool.
Arsenal have a limited transfer budget to play with this summer after missing out on next season’s Champions League.
Emirates Stadium bosses are ready to offer Tierney a long-term contract to make the move down south.
The Scottish Sun have claimed Arsenal are willing to pay Tierney more than £70,000-a-week, which equates to £4m annually.
Over a five-year deal, which would tie Tierney down in north London until 2024, he would receive a whopping £20m.
Arsenal have been slammed for the value of their opening offer for Tierney.
Former Scotland international James McFadden said: “[Virgil] van Dijk moves down for £13m plus his sell-on clause then goes for £75m. I think Kieran Tierney’s certainly at that level.
“We’re not talking about him going to a smaller side to build himself up. It’s one of the top sides in England coming and saying we’re going to take your best player for a pittance.
“Kieran Tierney is Celtic daft, so it’ll take something phenomenal for him to consider it, and also for Celtic to consider it.
“I think it would need to take someone who’s going to compete for a title in England or go and try to win the Champions League for Kieran Tierney to leave, because he’s winning trophies, he’s playing in Europe, he’s playing for the club he loves.
“I think it would have to be something really special, and Arsenal at the minute won’t challenge for the title. They might further down the line if they can invest in their squad, but £100m is nothing in England.
“I don’t think Arsenal will blow him away and make him think ‘I have to go and sign for Arsenal’.
“Arsenal look like they’re trying to do things on the cheap, and if it’s £15m it proves they’re trying to sign as many players as they can for as little as possible.
“It tells you that they’re not ready to challenge for the title, and that’s maybe in the back of Kieran Tierney’s mind to say ‘Why do I just want to go and try to finish sixth or break into the top four, when I could be winning titles for the club I love?’”