The Amsterdam club took 2600 fans to Glasgow for its Champions League defeat earlier this month.
But a section of supporters let themselves down by ripping up seats and flinging them at police and stewards.
A spokesman for Ajax said: "We have been advised the bill from Celtic will be £25,000 to replace the seats that were destroyed.
"We want to recover these costs from the fans guilty of this destruction. We hope that the footage will point out the perpetrators."