The Bhoys are due to face the Dutch giants in their next UEFA Champions League outing on Tuesday.
Boerrigter had been hoping to play a leading role in the contest, having completed a switch from the Amsterdam Arena to Parkhead over the summer.
He has, however, been laid low with an ankle injury and will find himself stuck on the sidelines as a result.
Boerrigter told the Daily Record: "I am sick at having to miss the Ajax game. I would have loved to play in it.
"Unfortunately, a small piece of bone in my ankle is damaged.
"It is a real shame, of all the matches for me to miss it would have to be the one against Ajax."
Celtic manager Neil Lennon added: "Boerrigter will take another week or so, he has a bit of soreness in the (ankle) bone and it can take a considerable period of time to heal properly.
"But we are hoping in a week to 10 days he will be able to join the rest of the players."