Genesio has come under fire in recent weeks after an inconsistent start to the season with OL eighth in Ligue 1 after eight games.
They have lost four times so far and are three points behind third-placed Marseille ahead of Friday's clash with reigning champions Monaco.
Aulas is not rushing to replace Genesio but he is set to assess his position following the St Etienne clash on November 5.
He said: "We'll assess the position after the derby. All coaches are in danger and we're an ambitious club. If I have to change coach then that won't bother me.
"If the results are better then we'll keep working together. If we don't manage that, then we'll do something else.
"There's no urgency to the situation. We're six points behind Monaco and we host them in our next game. I'm going to tell you that come the end of the season we are going to be where we want to be - with or without Bruno - because we had a really good transfer window and we've got a super stadium."