The 25-year-old has long been linked with a move to Arsenal, with the Gunners having had a bid turned down for the player in the summer.
However, Aulas has told the Gunners that they would be wasting their time by submitting a second offer.
"The journalists writing that Gourcuff is on his way out are simply getting it wrong," said the president.
"He will be staying. I brought him here because I thought that with him in our ranks, we would have a side with added value, especially in the Champions League."
Gourcuff has been capped 28 times by the French national team.