The Ligue 1 club are interested in signing the France striker if the Gunners buy Alexandre Lacazette.
But Olympique Lyonnais chief Jean-Michel Aulas has revealed there has been no offer from Arsenal yet – and the price will be at least £57m with Atletico Madrid still interested.
Giroud lost his starting place as centre-forward at the Emirates last year – and now risks slipping further down the pecking order.
Aulas told L’Equipe: “There has been an approach to Giroud. He is a boy we like, who also scores a lot of goals and who has a lot of ambition for Russia (next year’s World Cup).
“He told me himself that he wanted to take his time, that thing we not completely clear with Arenal, and he had the secret hope to be the No 1 attacker at Arsenal next season.”
Lacazette had been lined up to go to Atletico Madrid but the Spanish club has been banned from transfers this summer.
Aulas added: “I don’t have the intention to sell Alexandre quickly because there is still the will of Atletico Madrid to return on January 1. Now, I will listen also to what Alexandre says.
“I saw Arsene (Wenger) during the France-England game (June 13). I was with Gerard Houllier (the former Liverpool boss is an advisor to Aulas) and I seemed to understand that he said to Gerard that there could be interest from Arsenal.
“But it is premature to say that that things will happen. We need to get in a big transfer first. We cannot let a player of this quality leave without a replacement.”
Lyon are close to signing Bertrand Traore for £17million and are in talks to bring in Mariano Diaz from Real Madrid.
Asked if his club will sell Lacazette for €50-60million, Aulas responded: “It will have to be at least that.
“Everyone knows that Atletico’s offer was €53million plus €12m in bonuses. That makes €65million (£57m). As the Arsenal directors well know, it will be in that range.
“The price for a player who scores 25-30 goals every year – sometimes more than 30 – is at least that.
“But frankly, I haven’t yet thought about Arsenal for now because I thought that Alexandre was going to stay because the deal with Atletico fell through.”