The video was found on a mobile phone belonging to one of the passengers killed on the flight when German pilot Andreas Lubitz deliberately flew into a mountain, said the paper.
The scenes seen on the video were chaotic and very wobbly.
Prosecutor Brice Robin, who is handling the case in France, said none of the mobile telephones collected at the crash site had been sent for analysis.
"All are for now being kept at Seynes-Les-Alpes. If people at the site have picked up mobile phones, I am not aware of it," he told reporters by telephone. France's BEA investigation authority could not immediately be reached for comment.
On the video,  a banging of metal could be heard at least three times, possibly the sound of the pilot who had been locked out of the cockpit by Lubitz trying to break through the door.
Near the end there was a heavy shake and the cabin tilted sharply to one side. After further screams the video ended, said the paper.
The footage appeared to have been taken from near the back of the plane but no individuals could be identified.
A French magazine also ran a story on the video and printed an account of a conversation between the two pilots, according to a "special investigator".

When the captain left the cockpit to go to the toilet, he told Lubitz that he was in control. "I hope so", Lubitz replied, according to the magazine.
Later the captain implored Lubitz to let him in.
Lufthansa said on Tuesday that Lubitz had told officials at the airline's training school in 2009 that he had gone through  a period of severe depression, raising questions about screening process for pilots. Prosecutors have said he suffered from "suicidal tendencies" before obtaining his pilot's license.
Lufthansa is facing legal action from relatives of the victims and Chief Executive Carsten Spohr will be at the crash site in France on Wednesday.

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