The 31-year-old has been with the Gunners for more than five years but continues to find Premier League chances hard to come by and has made just one league start for the side so far this season, with his further 14 appearances all coming from the bench.
The Frenchman was close to securing a move to Everton over the summer but the deal fell through at the 11th hour and with the transfer window now back open, Giroud has again been linked with the Toffees, as well as Premier League rivals West Ham United.
According to Fanatik, however, Fenerbahce manager Aykut Kocaman is keen to bring in a new forward this month as replacement for former Arsenal man Robin van Persie and sees Giroud as an ideal candidate.
The publication claims that the club will try to "use the charm of Istanbul to lure him to Turkey" as well as the influence of his international teammate Mathieu Valbuena in order to get the deal done.
Giroud is currently on the sidelines with a knee injury but is expected to be fit again before the transfer window closes.