A hernia issue restricted Tierney to just 21 league appearances for Celtic last season, but the money Arsenal are willing to offer would make him the most expensive player in Scottish football history.
The Gunners have already failed with a bid of £15 million and have come back with a substantially improved offer. The £25 million comes into The Gunners £40 million transfer budget as they look to force their way back into a top four Champions League place.
Unai Emery is desperate to strengthen his defence ahead of the new season and left-back has been identified as a key area of concern amid on-going doubts over Nacho Monreal and Sead Kolasinac. With no other left back at the club, if Tierney moves, Celtic may well dip back into the market for another left back as cover.
Celtic rejected an opening bid of £15m last month but this latest development will further test their resolve.
Over the weekend, Celtic boss Neil Lennon had said that Tierney was struggling with a pelvic injury.
"KT is still having problems with the pubic injury, so we will need to get to the bottom of that", Lennon said.
Brendan Rodgers surprisingly chose to leave Celtic in February of this year in order to take charge of Premier League side Leicester City.
Celtic have lost players in the past that fans felt were not replaceable and the club has always moved on.
It would be a mega £97m double swoop if Arsenal could sign Tierney and Pepe, however, some British media outlet have dismissed the reports coming out of France and insist we haven't made a bid for the Lille forward so we'll have to wait and see how things develop.