The 31-year-old will consider his position at the Emirates in January and could leave to boost his chances of being selected in France's World Cup squad.
Everton and West Ham are both monitoring the situation and could make a move if Giroud decided to call time on his Gunners career.
However, Arsene Wenger insists his back-up striker will not be sold when the transfer window reopens.
The ‘Independent’ claim that Wenger will offer Giroud more Premier League minutes in an attempt to keep him at the club.