They were assaulted in the city square while enjoying the build-up to Scotland's Euro 2008 qualifying match against Ukraine.
The incident took place at Independence Square on Tuesday evening, according to the Scottish Football Association.
Some of the fans were treated for broken bones and cuts following the attack.
After the "entirely unexpected" incident, the Scottish FA has advised fans to be careful in the city, before and after the match and to avoid further provocation.
SFA security adviser Derek Kirkwood said: "The Scotland fans were absolutely magnificent, in that they did not react to the violence or provocation.
"They chose to walk away from the scene rather than get involved and their behaviour was exemplary."
Almost 3,000 Scotland fans are expected to attend the match.
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