Celtic chairman Brian Quinn has urged fans not to indulge in hateful singing that could see the club punished.
Quinn was full of praise for the behaviour of Celtic fans
Uefa will impose a minimum fine of £13,000 if supporters are responsible for racist or discriminatory conduct.
In an open letter to Celtic fans, Quinn said: "Offensive songs and chants have all but disappeared from Celtic Park.
"And we are working with the SPL, the SPL clubs and the police to discourage the remaining cases of unwelcome chants and songs at away grounds."
The appeal from Quinn follows a similar plea from Rangers chairman David Murray to Ibrox supporters last week.
Rangers were fined £13,300 by Uefa last season as a result of discriminatory chanting at their Champions League games against Villarreal.
"These (songs) can only damage the club and are a very real danger, as Uefa has made it clear it will treat this matter extremely seriously in the season ahead," added Quinn.
The behaviour of teams and of supporters inside and outside stadiums, is coming under scrutiny as never before
"Your behaviour has made us welcome at football grounds and cities around the world.
"This is something to be valued and protected at a time when those charged with running the game are likely to be especially vigilant."
Uefa have also warned they are willing to impose additional sanctions, such as forcing teams to play behind closed doors, closing grounds, awarding matches by default, deducting points and disqualification from competitions.
The provisions are in line with the stipulations introduced by the world governing body Fifa earlier this year.
The Fifa measures would see clubs deducted points in their domestic leagues if the problem continued.
Quinn continued: "It is clear that the football authorities are determined to do all they can to avoid further damage to the image of the game.
"The behaviour of teams on the field, and of supporters inside and outside stadiums, is coming under scrutiny as never before.
"As we saw last season, clubs are more likely than ever to incur heavy fines and to have points deducted for what is deemed to be unacceptable behaviour by their players and fans."