French prosecutors have formally opened an investigation into the armed robbery in Paris of Kim Kardashian in which the US reality TV star lost millions in jewellery, a judicial source said Wednesday.

No suspect has been arrested, but investigators are examining DNA traces found in the luxury apartment where five gunmen robbed Kardashian of jewellery worth some nine million euros ($10 million), sources close to the investigation said.

The star's 30,000-euro ($33,000) diamond necklace, found in the street by a passer-by hours after the October 3 heist, has not yielded any usable DNA samples, one of the sources told AFP.

Two mobile phones belonging to the 35-year-old multi-millionaire were also stolen during the holdup in the chic Madeleine district of Paris, near the city's main department stores and foreign embassies.

Police said the robbers held a gun to Kardashian's head, bound and gagged her and shut her into a bathroom in the biggest jewellery robbery of an individual in France in more than 20 years.

The star's bodyguard, Pascal Duvier, was not with her at the time because he was providing security for Kardashian's sister Kourtney at a Paris nightclub.

The robbers are believed to have fled the scene by bicycle.

Kardashian, who had been attending Paris Fashion Week, was briefly questioned by police before flying back to New York.

The mother of two and wife of US rap star Kanye West has kept a low profile since returning home to the United States hours after the robbery, which a spokeswoman said had left her "badly shaken".

Notably, her normally prolific Twitter account has fallen silent.

But on Tuesday she filed a lawsuit against US gossip website Mediatakeout.com after it alleged the robbery was a hoax designed to cheat her insurers out of millions.

The website "claimed, without any factual support whatsoever, that Kardashian faked the robbery, lied about the violent assault, and then filed a fraudulent claim with her insurance company to bilk her carrier out of millions of dollars," the lawsuit said.

FBI agents in New York may question Kardashian and members of her entourage who were in Paris at the time of the robbery, one source close to the French investigation said.

Kardashian has been a fixture of US celebrity news for more than a decade, known mainly for the TV show "Keeping Up with the Kardashians" and a leaked sex tape.

She has earned a total of some $131 million in the past three years, according to Forbes magazine.