The French striker only signed a new long-term contract extension in January, yet a lack of regular starts during the 2016-17 campaign combined with the club-record signing of prolific compatriot Alexandre Lacazette has led to sustained speculation that his five-year stint at the Emirates Stadium will come to an end during the current transfer window.
Giroud was included in Arsene Wenger's 25-man squad for their trip to Australia and China this week, giving rise to a belief that he would remain with the club.
However, speaking from Sydney, where Arsenal begin their pre-season campaign against Sydney FC at ANZ Stadium later on Thursday morning (13 July), the 30-year-old conceded that he might still move on.
"It (leaving) is an option to be honest with you," he was quoted as saying by The Mail. "I'm here in pre-season, I wanted to travel.
"Some people asked me if I am going to travel or not and of course, I am an Arsenal player and for the moment I am here so I just want to prepare well for the season and we will see later."
He added: "It is true that I really enjoyed my five years with Arsenal. I'm still an Arsenal player for the moment so I try to be professional. No matter what happens, I did my job on the pitch."
Giroud was keen to stress that he could yet prolong his stay at Arsenal and appeared to suggest that the extra competition provided by the arrival of Lacazette would help to improve his level.
Everton have been heavily linked with Giroud as Ronald Koeman wastes preciously little time in radically strengthening his squad for a potential top-four push, with the former Montpellier frontman and Swansea City playmaker Gylfi Sigurdsson believed to be the big-spending Toffees' next priority target after they wrap up a deal for ex-Southampton right-back Cuco Martina.
West Ham are also said to be interested after co-chairman David Gold admitted their desperate desire to acquire two new forwards over the coming weeks. A return to France with Lyon could be another possibility, while it was recently claimed that Ligue 1 rivals Marseille were confident of signing Giroud for a fee of approximately €28m (£24.7m, $31.9m).
It has also been claimed that Borussia Dortmund could move for him as a replacement for Pierre-Emerick Aubameyang.