The 22-year-old has been linked with the Gunners for some time, while Liverpool and Juventus are also said to be closely monitoring his situation in France.
Lyon are reluctant to part with the services of a key performer in their present and future, but would struggle to turn down a sizeable approach.
Grenier, who has been called up to the senior France squad for the first time, has revealed that he expects to be on his way to pastures new.
He has stopped short of revealing the identity of his next club, but admits a switch to the Premier League would appeal.
"I have more or less made a decision about which club I want to join," said Grenier.
"I have made the choice for sporting reasons above all. Arsenal? It could well be them.
"There would be competition for places, but you get that at all big clubs and it does not stop me making my decision.
"I am still young, but I have always set myself targets and I do everything possible to exceed them."