Giroud may be facing an extended spell on the sidelines after sustaining the damage in last weekend's 2-2 draw with Everton, just seconds after heading the equaliser.
The 27-year-old Frenchman has undergone two scans and is due to see a specialist on Wednesday who will provide a time frame for his rehabilitation.
Manager Arsene Wenger refused to speculate on the extent of the damage amid reports that Giroud has broken a bone and could be out for two to three months.
"Olivier will not play against Besiktas. He will see a specialist and afterwards we will make an announcement as to how long he'll be out," Wenger said.
"It can be (long term), we'll see. I don't know (if it's broken). It's sad for us, but for the moment we are focused on the game."
The loss of Giroud has robbed Arsenal of their first-choice striker as they seek to secure a 17th successive season in the group stage of the Champions League.
Wenger may now be forced into the transfer market in search of a replacement if the Gunners' worst fears are realised and Giroud faces an extended spell on the sidelines.
When asked if his absence would impact on transfers, Wenger replied: "I don't know. We are 24 hours from a big game and our focus is more on that than the transfer window.
"I've learnt one thing in our job - you say nothing categorically."
Summer signing Alexis Sanchez could operate as the lone striker against Besiktas, although doubts have been raised over his ability to perform the role.
Sanchez was substituted at half-time at Goodison Park following an ineffective 45 minutes as the focal point of Arsenal's attack.
Yaya Sanogo, Lukas Podolski and Joel Campbell are also options, although it is Sanogo, who missed out against Everton because of a hamstring injury, who is most likely to start.
Wenger, who ruled out Podolski leaving in the transfer window, is confident he has the options to cover for Giroud's absence in the second leg of the qualifier at the Emirates Stadium.
"Yes of course, why should I not have that confidence?" Wenger said.
"Nobody can image that when one player is missing, no matter how big the player is, the team can't win a game any more."
The first leg finished in a goalless draw in Istanbul and while Arsenal are heavy favourites to defeat the Turkish club, Wenger knows anything is possible.
"For us the Champions League is very important. We want to play at the top level, at international level," Wenger said.
"It's a very important game, more on the sporting side than the financial side.
"Nothing is unthinkable - it (not qualifying for the group stage) has happened to 99 per cent of all the big clubs in Europe.
"But we want to be in there and we think we have the potential to do it. The task is clear and we know we can do it. So let's do it.
"Experience plays a big part. We have big experience of playing in the Champions League and in highly pressurised games. We can use that.
"But what's more important is putting in a good, strong performance.
"I'm confident this group is mentally prepared and ready for the fight."