A deal between the two clubs for the 23-year-old centre-back has all but been agreed, with just the finer details to be sorted out, according to Sky Sports.
Ajer, who was linked with a host of clubs earlier this summer including the Bees’ fellow promoted side Norwich City, and Newcastle United, was omitted from the Hoops squad for their pre-season friendly at the weekend and was not named in the club’s European squad.
Celtic face Brentford's sister club Midtjylland of Denmark in tonight’s Champions League qualifiers.
The EPL new boys have already signed Frank Onyeka from the Danish side after the Nigerian midfielder received a work permit and are bolstering their squad ahead of their maiden tilt at the English top flight.
Ajer joined Celtic in February 2016 after playing more than 60 times for IK Start in Norway’s Tippeligaen, and after a brief loan spell with Kilmarnock in 2017, became a mainstay in the Bhoys defence, making 170 appearances and scoring six goals in all competitions.
HIs performances at the back for Celtic led to rumoured interest from Bayer Leverkusen, AC Milan, Zenit St Petersburg, Leicester, Tottenham Hotspur, Southampton, and Liverpool.
New Hoops boss Ange Postecoglou has already brought in Osaze Urhoghide from Sheffield Wednesday, who is expected to operate at the heart of defence, while the Scottish Premiership side has also been linked with moves for Croatian centre-back Marko Vuskovic and former Parkhead defender Erik Sviatchenko.
Speaking to the Daily Record recently, the former Yokohama F. Marinos boss insisted he was fully focused on the future.
He added: "The guys who are going to leave will leave; that’s not my department. I’m concentrating on what’s in front of me.
“[Ajer] wasn’t involved today. I’m concentrating on the ones who are involved.
“I can’t afford to waste time on anyone other than the ones who are here and committed to being here.
“It’s pretty clear we need to bring players in. That’s not an easy process but we are working hard, hopefully in the next week.”