The lawsuits were submitted on behalf of 28 Syrian refugees in Jordan who claim they were forced to leave Syria.
The human rights lawyers are calling on the ICC to investigate possible crimes against humanity committed since the Syrian revolution began in 2011.
The civil war has left more than 500,000 people dead and millions displaced.
Syria is not a party to the Rome Statute, the treaty which created the ICC, meaning that it has not been possible to bring an international criminal case against its government.
However, lawyers have used a recent ICC ruling on Rohingya Muslim refugees in Bangladesh to launch two lawsuits last week.
In September 2018, judges ruled that while Myanmar is not a member of the court, Bangladesh is, and that because part of the alleged crime happened on Bangladeshi territory the prosecutor does have jurisdiction.
Last week’s lawsuits were filed based on the same principle that the 28 Syrian claimants fled to Jordan, which does fall within the ICC’s jurisdiction.
Victim testimonials stated that they were tortured, bombed and shot at, which forms a substantial part of the evidence that has been submitted.
The first case was filed on Monday 25 February by the Guernica Centre for International Justice and the second was filed on Thursday 28 February by a team of British human rights lawyers.
Rodney Dixon QC, who is leading the legal team that filed last Thursday’s lawsuit, said the case represented “a genuine breakthrough for the Syrian victims”.
“There is a jurisdictional gateway that has opened up finally for the ICC prosecutor to investigate the perpetrators who are most responsible,” Mr Dixon added.
Previous attempts at prosecuting President Assad and members of his regime were unsuccessful because the ICC had not accepted it has jurisdiction over Syria.