That would leave Arsène Wenger facing a race against time to sign a new striker before the transfer window shuts next Monday.
Giroud sustained the injury in the dying seconds of the 2-2 draw at Everton after scoring Arsenal’s late equaliser.
Arsenal are hoping Giroud has not sustained a fracture, but there is major concern at the Emirates that there is a break, after the Frenchman was sent for two scans.
The club are mindful of the fact that a hairline fracture to Jack Wilshere’s foot only showed up on a second scan after the initial swelling had gone down.
Giroud is definitely out of the Champions League play-off second leg against Besiktas tomorrow night and the Premier League trip to Leicester on Sunday, and there is a real concern that he could be out for a prolonged spell.
Losing Giroud for any length of time would be a blow to Wenger and will force the Arsenal manager to rethink his transfer plans ahead of the window closing.
With top-class strikers hard to find during the final week of the transfer window, Wenger may revive his interest in Queens Park Rangers’ Loïc Rémy, or even consider a short-term deal for Samuel Eto’o.
Danny Welbeck has been told that he can leave Manchester United and could also be considered.
A clause in Rémy’s QPR contract means Arsenal could sign him for £8 million, but the Frenchman’s £90,000-a-week wage demands have so far put Wenger off.
More ambitious targets would be Paris St-Germain’s Edison Cavani or Monaco’s Radamel Falcao, but both men would be very expensive.
A season-long loan deal for Falcao would cost in the region of £20 million.
Lukas Podolski is seeking a new club, with Juventus and Wolfsburg both keen to sign the German.
Wenger could be more reluctant to let him go if Giroud is out for a while, although the club could also put the £11 million fee and £100,000-a-week wages towards Rémy or a different target.
While Giroud waits on his scan results, Wenger has to plan for the Besiktas game without the France international, Mikel Arteta and suspended Aaron Ramsey.
Arteta sustained a sprained ankle in the first leg of the Besiktas tie, but he is expected to be back for the Manchester City game on Sept 13.
A successful passage into the group stages of the Champions League is likely to see Wenger attempt to sign a defensive midfielder, despite the good news on Arteta.
Arsenal are understood to have been in talks with Sporting Lisbon over William Carvalho, with a bid lined up for this week.
Sporting value Carvalho at around £28 million and are keen to secure players on loan from Arsenal as part of any deal.
Arsenal have already told winger Serge Gnabry that he can leave on loan and the club would not stand in the way of Francis Coquelin moving to Portugal.
Having seemingly lost out to Roma in their attempt to sign central defender Kostas Manolas, Arsenal may turn their attention to West Ham United’s Winston Reid.