The Frenchman has enjoyed a fine season at Lyon and his performances have attracted the attention of Gunners boss Arsene Wenger, who is looking to spend big this summerLyon president Jean-Michel Aulas has warned Arsenal they will have to pay £31 million in order to sign Clement Grenier this summer.
"Clement said to me that he will give priority to Lyon if we qualify for the Champions League. So I prefer [to listen to] what he said, rather than what Arsene Wenger said," Aulas told Canal+.
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"If Arsene wants to have an idea of Grenier's price, I will tell him that it is the same as what Bayern Munich paid for Mario Gotze - around €37m (£31m)."
The Gunners are keen to strengthen their squad this summer after securing Champions League football ahead of local rivals Tottenham on the final day of the season.
There has been talk of Wenger having up to £70 million to spend with the likes of Steven Jovetic, Wayne Rooney, Sebastien Corchia as well as a potential new goalkeeper all on the radar.
Goal revealed earlier this week that Swansea defender Ashley Williams could well be the first new arrival at the Emirates Stadium, with the 28-year-old already having agreed terms with the Gunners.
However, Wenger has previously not spoken publicly on any potential transfer target, before his claims to Telefoot that he was "following" Grenier.
The 22-year-old has enjoyed a breakthrough season for his current club, cementing himself as a first-team regular and scoring six goals in 32 appearances in all competitions.