It is understood that Celtic have agreed a fee with FK Start for the 17-year-old to join the Glasgow club in the summer.
The midfielder trained with the Scottish champions for a week in January before returning home to Norway.
However, he is flying to Scotland on Tuesday to finalise terms and complete a medical.
Ajer, who is in La Manga, Spain, where he played for Norway under-19s against Portugal on Monday, was quoted as telling Faedrelandsvennen: “Yes, I travel down tomorrow and the clubs agree.
“So we will look at my appointment and the medical test. It is amazingly big for me.”
THE SCOTSMAN ON SOCIAL MEDIA