The France international has endured a tough time at Stade Gerland, with injuries preventing him from making the desired impact.
A fresh start has been mooted for some time, and Arsenal are regularly touted as the ideal club to help Gourcuff get his career back on track.
Arsene Wenger has admitted to being an admirer of the talented playmaker in the past, but no deal has been discussed and that remains the case during the current transfer window.
Lyon chairman Jean-Michel Aulas told BeIN Sport: "Arsenal and Gourcuff? I don't know about any offers.
"I have never spoken with Arsenal."
With Lyon having previously revealed that they would be willing to part with key men in January in an effort to free up funds, various other players have attracted interest from afar.
On reports linking Lisandro Lopez with a switch to Serie A giants Juventus, Aulas said: "There was an offer but it didn't match our valuation or the desire of the player. We can forget about that."
On speculation suggesting Jimmy Briand could be off to Turkish side Besiktas, he added: "We have accepted a proposition for Briand, but all three parties now need to agree."