The man is believed to be Maxime Hauchard from a small village in Normandy in northern France.
Earlier, French interior minister Bernard Cazeneuve had told reporters that the Frenchman in the video had "left for Syria in August 2013 after a stay in Mauritania in 2012."
The prosecutor's office said "circumstantial evidence confirms the involvement of a Frenchman in the decapitation of Syrian prisoners shown in an Islamic State video released on Sunday."
An investigation into Hauchard was opened in August by French authorities "for criminal association in relation to a terrorist organisation," a judicial source said.
The video released on Sunday shows the young, bearded man in camouflage fatigues alongside fellow combatants.
Prisoners are lined up on their knees in front of them. The man resembling Hauchard is seen placing a knife to the collar of one of the prisoners. The execution itself is not shown, but the severed head is later seen.
An intelligence source said it was being verified whether a second French citizen was among the jihadists seen in the video.
Around a thousand French are thought to have taken part in the conflict in Syria and Iraq, with around 375 currently there, the government has said. At least 36 have died there.