M6, the French TV channel, reported that the pilot then initiated a 'rapid descent' aimed at putting out the fire on board, and clearing the smoke.
Rapid descents involve dramatic changes in cabin air pressure, and can be extremely dangerous, but the claims about the flight's last moments fit in with earlier information.
According to Greece's defence minister, Pano Kammenos, the plane dropped sharply from 37,000 feet to 15,000 feet, and then made 'sudden swerves'.
As it entered Egyptian airspace, over the Greek island of Karpathos, the first turn was a sharp, 90-degree one to the east, and then there was a full circular loop.
A leaked data report also suggests that a fire blazed across the flight deck minutes before disaster -suggesting a catastrophic electronics malfunction.
The new information made terrorism seem 'less likely', although it has still not been ruled out.
On Friday, authorities released an audio recording of Captain Shoukair's words to Swiss air traffic control, from around an hour into the planned four-hour flight.
The communication occurred around midnight local Swiss time, about two-and-a-half hours before Greek air traffic controllers in Athens lost contact.
These messages were sent to ground computers before the airliner plunged from the sky, killing ten crew and 56 passengers, including one Briton.
The hunt is now on for the plane's black boxes - the flight data and cockpit voice recorders that could unlock the mystery.
French foreign minister Jean-Marc Ayrault said 'all theories are being examined and none is favoured'.
But Simon Hradecky, editor of the highly respected website Aviation Herald, said available data suggested an electrical fault on the jet was more likely than a terrorist attack.
Details of Flight 804's final moments emerged as human remains and personal belongings were over the weekend recovered by search vessels from some of the victims who died.
The Egyptian military released images of some items found, including life vests, parts of seats and objects clearly marked EgyptAir.
Debris was found 130km north of the port of Alexandria by the Egyptian navy. The spot is south of where the Airbus vanished from radar signals.
