Gunners boss Arsene Wenger has already confirmed his interest in 22-year-old Grenier and he wants to add him to his ranks at the Emirates Stadium in this summer's transfer window.
However, Lyon boss and former Arsenal player Remi Garde recently issued a hands-off warning and claims the player would be best off staying where he is.
And, while Grenier has been valued at around 35 million euros by Lyon, which would be prohibitive for the Gunners, the player has not ruled out moving to the north Londoners.
Speaking while on duty with the France national team, Grenier told Le Progres: "Have I made a decision? Yes. More or less. I keep it for myself as I care for my discussions I have had with the president and the coach.
"I just want to say that I've never had demands. Moreover, either with the club or with my advisors, figures were never mentioned. My choice is a sporting one.
"A fight between Arsenal and Lyon for me? Maybe. The competition at Arsenal? There is already that here [at Lyon].
"It is the same in all the major clubs and it is not a hindrance in my decision making. I'm still a young player. But I always set goals and do everything to surpass myself. Whether it is at Lyon or elsewhere.
"For now, I'm still a Lyon player and I may stay next season. I do not know myself what will happen. The only thing I expect is really at a sporting level, and my desire is to progress."