Mourinho has been out of work since December last year, when he was sacked by Manchester United amid a run of poor results.
The Portuguese coach has this season made regular television appearances as a pundit, widely attracting acclaim for his insight, demeanour and analysis.
Lyon sacked Sylvinho after just one win in his last nine matches at the helm and Aulas had identified Mourinho as a target, though the former Chelsea, Inter and Real Madrid boss apparently turned them down on account of having another job lined up.
"We had some fairly flattering exchanges over text," Aulas told reporters via a telephone news conference.
"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.
Aulas confirmed the decision to dismiss the former Manchester City and Barcelona defender came from Juninho, with Sylvinho's perceived negative brand of football doing little to endear himself.
"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 said.