The Syrian opposition say one of those attacks in August 2013 on the rebel-held suburbs of Damascus killed around 1,400 people, including more than 400 children, in one of the many horrors of the 13-year civil war.
Prosecutors said they had made the request to the Paris Court of Cassation on Friday on judicial grounds, two days after another appeals court upheld the arrest order issued in November.
"This decision is by no means political. It is about having a legal question resolved," the prosecutor's office at the court said in a statement.
France is believed to have been the first country to issue an arrest warrant for a sitting foreign head of state.
Investigative magistrates specialised in cases of crimes against humanity issued the warrant after several rights groups filed a complaint against Assad for his role in the chain of command for the alleged chemical attacks on August 4, 5 and 21, 2013.
But prosecutors from a unit specialising in investigating "terrorist" attacks have sought to annul it, although they do not question the grounds for such an arrest.
They argue that immunity for foreign heads of state should only be lifted for international prosecutions, such as ones at the International Criminal Court (ICC).
Alongside Assad, the warrants target his brother Maher -- the de-facto head of the Syrian army's elite fourth division -- and two generals, Ghassan Abbas and Bassam al-Hassan.
The anti-terror prosecutors contested only the warrant for Bashar al-Assad's arrest.
'Political manoeuvre'
The Syrian Centre for Media and Freedom of Expression (SCM), lawyers' association Open Society Justice Initiative (OSJI) and the Syrian Archive, an organisation documenting human rights violations in Syria, filed the initial complaint.
SCM head Mazen Darwish criticised Tuesday's move.
"We view (the) filing of the appeal as a political manoeuvre aimed at protecting dictators and war criminals," he told AFP.
Lawyers Jeanne Sulzer and Clemence Witt, who are representing the plaintiffs, said the appeal to the Court of Cassation "again threatens the efforts of victims to have Bashar al-Assad judged in an independent jurisdiction".
A UN report on the August 21 attacks said there was clear evidence sarin gas was used in Moadamiyet al-Sham as well as Zamalka and Ein Tarma in the Ghouta suburbs of Damascus.
Syria's civil war has killed more than half a million people and displaced millions since it broke out in March 2011 with the Damascus authorities' brutal repression of anti-government protests.
Syria in 2013 agreed to join the Organisation for the Prohibition of Chemical Weapons shortly after the Ghouta attacks.
But the global watchdog later accused the Assad government of continuing to attack civilians with chemical weapons in the war, charges Damascus denies.
Syria's OPCW voting rights were suspended in 2021, in an unprecedented rebuke following poison gas attacks on civilians in 2017.
The OPCW found that the Islamic State jihadist group, another actor in the Syrian conflict, used mustard agent in a 2015 attack in northern Syria.