Gunners boss Arsene Wenger is a long-term admirer of pass-master Grenier, who has enjoyed a superb campaign in the Lyon midfield.
And after revealing that he will depart the French club when the transfer window opens, the 22-year-old says he may well be tipping up at the Emirates Stadium.
“I have made a decision about which club I will join this summer,’ said Grenier on French television.
“I have made the choice for sporting reasons above all.
“Is it Arsenal? It could be them.
“There is competition for places there, but you get that at all big clubs. It won’t stop me making my decision.
“I’m still young, but I have always set myself targets and I’ll do everything possible to exceed them.”
Grenier scored seven goals in 32 matches for Lyon last season whilst also picking up nine assists, and has drawn comparisons to Real Madrid ace Kaka.
Manchester United were previously interested in his services, but those links have since cooled.
Arsenal are also believed to be keen on Grenier’s Lyon team-mate Maxime Gonalons, having scouted him intensively in recent months.