Atar helped coordinate the attacks from Syria, although a number of other organisers have yet to be identified, Le Monde newspaper reports.
Atar is believed to have recruited the two Iraqis who blew themselves up outside the Stade de France football ground on 13 November 2015, to have organised their voyage to France and to have received the plans for the Brussels attacks the next March.
Two other men, Algerian Adel Haddadi and Pakistani Mohamed Usman, who travelled with the Iraqis but were detained in Austria for carrying false passports, identified Atar as being "Abou Ahmad" from photos.
Family involved in attacks
Two of Atar's cousins, Ibrahim and Khalid El Bakraoui, organised the logistics of the Paris attacks and blew themselves up during the Brussels attacks.
His younger brother, Yassine Atar, was arrested shortly before the Brussels attacks in possession of a key to the apartment that the plotters used as a hideout.
Met Islamic State founders in US-run prison
Atar himself was jailed by an Iraqi court in 2005, serving time in the US-run prisons at Abu Ghraib, Camp Cropper and Camp Bucca.
Camp Bucca gained a reputation as an incubator for the Islamic State (IS) armed group, whose future leader, Abu Bakr al-Baghdadi, was imprisoned there.
Atar was freed on humanitarian grounds in 2012 after a campaign that claimed he had cancer of the kidney.
He subsequently went to join IS in the Syrian town of Raqqa, where he recruited and briefed the Paris bombers and the two would-be attackers who were arrested in Austria.
Computer reveals plans for more French attacks
According to data on a computer found by Belgian police in a dustbin outside the Brussels attackers' hideout, the group had compiled a list of targets in France, including Paris's La Défence business district, but dropped the plan and launched the Brussels attacks instead.
The computer also gave investigators a number of clues allowing them to identify Atar.
To read our coverage of the Paris attacks click here