The newly-promoted English Premier League club have been in talks over Ajer in recent days, with an agreement now thought to have been reached.
Reports in his native Norway claim the 23-year-old centre-back is set to undergo a medical and that personal terms are also in place.
Celtic are close to selling centre-back Kristoffer Ajer to Brentford in a £13.5million deal
Reports in Norway suggest he is set to undergo a medical and personal terms are also in place
Ajer went public on his wish to leave Parkhead at the start of last month and had attracted interest from Newcastle United, Norwich City and Bayer Leverkusen.
Celtic rejected a £10m bid from Norwich but are now poised to bank a sizeable sum for a player in the final year of his contract.
If the transfer is concluded, it would provide a significant boost to the funds available to Ange Postecoglou's squad rebuild this summer.
The Australian is on the brink of landing £4.2m defender Carl Starfelt from Rubin Kazan as Ajer's replacement, having already secured Israeli winger Liel Abada in £3.4m move from Maccabi Petah Tikva.Insurance Loans Mortgage Attorney Credit Lawyer
The newly-promoted Premier League club have been in talks over Ajer in recent days
It is still unclear, however, whether Starfelt will be able to play a part in either leg of Celtic's forthcoming Champions League qualifier against FC Midtjylland.
Right-back is another priority