The 24-year-old appeared on the verge of joining the Premier League side as the clubs discussed a £60 million (€68m/$80m) transfer, only for Lyon to declare that he is going nowhere when negotiations hit a snag.
Aulas revealed that Lyon called off the deal because it was taking too much time and added that there is further interest in the France star, with Real Madrid among those tracking him.
But despite the offers coming in, Aulas is confident the Ligue 1 side can hold onto their attacking midfield star.
"I had an exchange with him the day after we decided to stop the negotiation with Liverpool. He was very serene," Aulas told Le Parisien.
"I immediately told him that if he wanted to stay, we could extend. I had said before that we did not need to sell. We have the means to keep Fekir.
"The current trend is that we do not let anyone go to have the best team and that we will strengthen it as planned."
And Aulas says he has already heard from the France Under 21 captain as they turn their attention to other targets.
"He called me to tell me he was not coming. We cannot do anything with young players," he added.
"In principle, we look for an older defender. We had an opportunity, but it is necessary that the salary of a player in a French team is not incoherent with the other players of the same status."