The France forward signed a new contract until 2020 this season and started only 11 Premier League games in the last campaign as Alexis Sanchez was deployed as centre-forward.
Arsene Wenger is looking to further strengthen the striking department this summer.
And speaking before Les Bleus World Cup qualifier against Sweden, Giroud said: “I will put up with the transfer market. I will discuss with the coach but he is relying on me.
“It is true that I have had limited playing time but it was for a certain amount of time, it is true there were things against me. I will not settle for another year with such little playing time.
“I have will have to think carefully to consider it with my relatives and my advisers. It will be a decision that will be maturely thought about to have more more playing time.”
A move back to Ligue 1 could be on the cards with Olympique de Marseille thought to be interested and when quizzed on the prospect of moving to the south of France, Giroud fuelled speculation by revealing his admiration for the club.
“I’m not necessarily insensible to the approaches of French clubs,” he added. “Especially a legendary club like Marseille.”