Asbestos was widely used as a building material and manufacturing before being banned in France in 1997.
The French Senate says that over three decades 30,000 people have died from inhaling asbestos.
The Aubry scandal dates back to the 1980s.
Aubry was responsible for implementing European Union directives regarding the presence of asbestos in workplaces.
She's accused of taking too long to enact the rulings, thus putting workers' lives at risk.
Judges want to establish whether Aubry's decisions were swayed by an industrial lobby called the permanent asbestos committee.
Yesterday afternoon hundreds of protesters gathered outside Paris's central courthouse, calling on magistrates not to drop the investigation.
Victims’ groups insist the culprits must be brought to justice.
"I don't think it would be right for there to be impunity for the poisoners,” said Eric Halgand a member of a victims’ group on the demonstration.”I think the judicial authorities are taking too long over this, but I understand why, because the people involved are in top positions, and when this all comes out it's going to hurt."
On Thursday right-wing French daily Le Figaro reported that Aubry is unlikely to be charged.
Judicial sources said that the criminal investigation would probably be dropped.
That would leave her free to bid for a seat in the current Socialist government, although she has a troubled relationship with President François Hollande, having stood against him in the primary to become the party’s presidential candidate last year on a slightly more left-wing platform.