The journalists on site reported that Mr Sarkozy left around 30 minutes after he arrived at the clinic in order to fly to Frankfurt to an “informal meeting” with Chancellor Angela Merkel.
The neighborhood in the French capital’s 16th arrondissement will soon calm down since the birth of the first lady is imminent.
According to a police source cited by the BBC, doubling the security around the clinic has been decided on Tuesday night in preparation for the visit of Carla Bruni-Sarkozy.
The Italian-born first lady has reportedly reserved four suites, each valued at 250 euros per night: one for her, another for the father and two for the bodyguards.
Hospital workers also told a French radio station that the suites are not “particularly luxurious”.
French newspaper Le Figaro claims that the baby will be a girl, but the Elysee Palace has never publicly commented on the pregnancy, which 56-year-old Mr Sarkozy and his wife regard as a private matter.
The Elysee has already warned that no official statement would be issued after the delivery.
Mrs Bruni-Sarkozy has said that the French public has little interest in the pregnancy and that she was “in a hurry to get it over with”.
The former model already has a 10-year-old son, Aurelien, with the philosopher Rafael Enthoven.
President Sarkozy has two sons, Pierre, 26, and Jean, 25, from his first marriage to Marie-Dominique Culioli, and a third, Louis, 14, from his second to Cecilia Ciganer-Albeniz.