The ICC, founded in 2002, is the world's only permanent war crimes court. But moves so far to refer Syria to The Hague-based body have been unsuccessful as Russia has blocked them with its veto in the UN Security Council.
The UN go-ahead is needed as Syria is not a member state.
Ayrault told AFP that action "can be launched it they concern (French) nationals who are engaged in the war in Syria alongside Daesh," using another name for the Islamic State group.
Russia dealt the International Criminal Court a blow Wednesday saying it was formally withdrawing its signature to the tribunal's founding Rome Statute.
"The court did not live up to the hopes associated with it and did not become truly independent," Russia's foreign ministry said, describing its work as "one-sided and inefficient".
Ayrault said the "action is symbolic and political which I regret because it would have been preferable that Russia does the opposite" and ratifies the founding Rome Statute.
In May 2014, France presented a draft resolution calling for war crimes to be investigated in Syria, but the measure was defeated when Russia and China vetoed the request.
A country that has signed up to the Rome treaty or whose citizens have been the victims of crimes may refer cases to the ICC's chief prosecutor for investigation.
Cases may also be referred by the UN Security Council, and the prosecutor can initiate her own investigations with permission from the judges providing member states are involved, or a non-member state can agree to accept the court's jurisdiction.
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