The Gunners made a £6m approach for the 22-year-old over the summer, but the player himself opted against completing a switch to the Emirates Stadium to boost his development.
But that has not deterred Arsene Wenger, with the Frenchman still monitoring his compatriot.
Grenier has been one of Ligue 1’s standout performers in recent times in an underperforming Lyon side – who currently occupy a midtable position.
The Stade Gerland-based side are currently struggling to compete with the likes of Paris Saint-Germain and Monaco financially, which has led to rumours that they are willing to sell Grenier in the New Year.
Despite having a wealth of central options, Wenger is keen to take advantage of the situation to lure the youngster to London.
Serie A giants Juventus are also interested, but Arsenal’s reputation for nurturing talented young players may give them an advantage.
Grenier admitted recently that, although his contract runs until 2016, he may leave Lyon soon.
“I have a contract which now runs until 2016. We’ll see what happens here in a year. Anyway, I told the President that whatever happens, I will not leave my club for free,” Grenier said.
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