Aulas retweeted a message from a fan criticising Fournier for starting Yoann Gourcuff on the bench during Lyon's 2-1 loss at Lille.
Fournier said Aulas was entitled to his opinions, but he would continue to select his team as he saw fit.
"We sometimes have differences or discussions. Then, what is unfortunate is to put it on social network. It's part of the president's communication," he told a media conference on Friday.
"I do not have to go into it. He is entitled to his opinion. After that, it's me, that until further notice will decide until the end of the season.
"Then, we will review our season we have spent in working together. For now, it is I who decides. I will try to put the best possible starting line up to go get a positive result in Montpellier."
Gourcuff could be set to start for the league leaders when they travel to Montpellier on Sunday.
Fournier said his treatment of the France international was no different to every other player in his squad.
"I manage him like I do that other players. He could start. He has started three of the last five matches," he said.
"We are counting on him a lot... it's my decision. I will try and pick the best XI for Montpellier.
"There is a balance to be struck between the talent and the merit. It's very important for the squad. And we can't overlook the need to perform on the weekend."
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