Aulas revealed Lyon turned down an offer for the France midfielder, who joined from Bordeaux in 2010.
"There was a request from Arsenal and I said no. He has a five-year contract, wants to succeed with Lyon and I believe him," Aulas was quoted as saying by L'Equipe.
Arsenal lost key midfielders Cesc Fabregas and Samir Nasri over the summer, but have since brought in Mikel Arteta from Everton.
Gourcuff, 25, who has also played for AC Milan, was troubled by injury in his first year at Lyon.