"The atmosphere in the Karaiskakis Stadium is boiling hot," said Moroccan Zairi. "Our supporters are fanatics and are literally our 12th player.
"Such support can turn a match. We have already beaten Arsenal at home this season and we will give everything to win on Tuesday."
However the Greeks, who are making only their third appearance in the knockout rounds, are up against one of the Champions League's form teams.
Laurent Blanc's swashbuckling side are top of Ligue 1 and dropped just two points in the group stages of Europe's elite club competition, conceding only two goals.
"Bordeaux are a quality side with a lot of talented, well-known players and an exceptional coach in Laurent Blanc," he said.
"They cause teams all kinds of problems. We have analysed a lot of their matches but in the end the main focus will be on our own approach to the game."
Olympiakos, who are coached by Serbian Bozidar Bantovic, have an unblemished home record in this season's competition following 1-0 victories over AZ Alkmaar and Arsenal and a 2-1 win over Standard Liege.
"We must not fear the hot atmosphere Olympiakos fans can create," he said. "We have got our fans used to victory.
"We have no right to fail now. Our aim is to go through to the quarter-finals and it is not beyond us."
The game represents a perfect chance for Bordeaux to deflect attention away from the media speculation surrounding Blanc's future.
Blanc is widely considered the favourite to succeed Raymond Domenech as France coach after this year's World Cup and he said himself last week he did not know what he would do next season.
"If somebody is able to say what I'll do tomorrow he is very clever as I don't know myself," Blanc told the French paper L'Equipe.
