The Portuguese, who left his previous coaching position with Manchester United almost a year ago, has yet to return to the game in a major capacity since his Old Trafford exit.
Instead, the former Real Madrid boss has turned to punditry, making regular appearances on television while dropping hints about his future.
Ligue 1 heavyweights Lyon sacked Sylvinho on Monday after just one win in his last nine matches at the helm and Aulas had identified Mourinho as a target to take the helm.
However, speaking to reporters via a telephone news conference, the 70-year-old revealed that the ex-Chelsea and Inter man turned them down on account of having another job lined up.
"We had some fairly flattering exchanges over text," Aulas stated. "It was nice for everyone. He did not accept our proposal to meet because he has already chosen another club."
Sylvinho's arrival coincided with the return of club icon Juninho Pernambucano as sporting director in pre-season.
"It was indeed Juninho who took the decision to change coach, he even had this view a little earlier on," Aulas said. "He is going to continue to take responsibility on the football side in the same way.
"He is a boy who breathes football. In the first matches, he detected a certain number of problems. It is he who explained his decision to Sylvinho.
"I didn't know of him (Sylvinho) before Juninho told me about him. The defensive style he offered created a destabilising effect in terms of the normal attacking style of Lyon. We made a mistake, but in general we don't make a mistake twice."
While Aulas seemed to confirm Patrick Vieira will not be considered as he is under contract at Nice, Remi Garde – who coached Lyon between 2011 and 2014 – is an option.
"He hasn't been contacted yet, but that is a possible route. For the moment, talks haven't happened, but maybe they will," Aulas added.