The publication is reporting that three of the five men have been arrested, though it is unclear as to whether the jewels have been recovered.
No one was injured in the raid on the opulent hotel, in the 1st arrondissement of Paris, police have confirmed.
"The attackers then used axes to smash open reinforced glass cases containing jewelley, while filling bags", according to a source at the famous hotel.
She added: "One man wearing a ski mask ran from the bar door to the front of the bar right past my father".
Sky News reports that a witness also heard "gunfire" at the scene.
Writer Frédéric Beigbeder took refuge in a basement as the men - armed with an axe and two handguns - swooped on the hotel.
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TV station BFM said the jewels were worth €4.7m (£4.16m).
The Ritz-Carlton is located in the 1st arrondissement and overlooks the Place Vendôme.
In more recent history however, the Ritz has become known as the place where Britain's Princess Diana spent her last hours before a auto accident in a tunnel in the French capital while being pursued by paparazzi.
Mohammed al Fayed bought the hotel in 1979 and carried out major renovations to bring it back to its original popularity.
The Ritz was the last building Princess Diana and Mr al Fayed's son, Dodi, left before their vehicle journey that ended in tragedy under the Pont de l'Alma on 31 August 1997.