The clash between supporters and police occurred in Amsterdam before the UEFA Champions League meeting between the teams, with Ajax winning the encounter 1-0.
Police said mostly Celtic fans had been arrested but reports suggest supporters of other clubs were also involved.
"There were 15 arrests, mostly Celtic supporters," Dutch police said in a statement.
"At the end of the afternoon, a large group of Celtic supporters attacked police officers in plain clothes. Eight were injured and one left unconscious."
According to police, 'bottles and sticks' were used in the attack, with some officers suffering broken noses among other injuries.
Lasse Schöne's goal was all that separated the teams, leaving Celtic bottom of Group H and unlikely to reach the knockout stages.
Celtic fans were told to stick to certain areas ahead of the trip to the Netherlands after a violent incident in a city bar on Tuesday.
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