Update 10.20pm: The three suspects were arrested near the chic Place Vendome in central Paris but none of the jewels or other items has been recovered so far.
The robbers appeared to have entered the Reza jewellery store through a side street entrance, smashed display cases and taken the valuables. The store also opens into the five-star Hotel Ritz.
Two people were inside the store at the time and alerted police.
Several high-end Paris jewellery stores have been targets of dramatic robberies in recent years, including Cartier, Harry Winston and Chopard.
Kim Kardashian West lost millions of dollars worth of jewellery when she was robbed at gunpoint in a Paris apartment in October 2016.
The Ritz was an especially luxurious target. The 19th century hotel, which reopened in 2016 after a four-year renovation, has housed such famous names as Ernest Hemingway and Coco Chanel.
It was the last place Princess Diana stayed before her fatal car crash and is popular with elite guests from around the world.
Update 9pm: Armed robbers have stolen jewellery reportedly worth millions of pounds from the Ritz hotel in Paris.
French police said that five thieves, some of them armed, entered the luxury hotel and stole goods worth "an important sum".
The French TV station BFM said that €4.7m worth of jewels have been taken.
A police spokesman said that no one was injured in the robbery at about 6.30pm on Wednesday. He said that three robbers were arrested and the other two got away.
The officer would not confirm reports that jewels were stolen from a hotel shop or that the robbers were armed with hatchets.
- PA and Digital Desk