A police spokeswoman said the teenager seized the hostages in the town of Blagnac, near Toulouse airport, wearing a motorcycle helmet at around 4.15pm local time.
Officials said it appears to be a botched armed robbery and that the incident did not appear to be terrorism motivated. No injuries have been reported.
Local media said the suspect fired two shots at police when they tried to approach the shop.
Television reports said the hostage-taker has demanded to speak with a negotiator.
Le Parisien and Le Figaro have both quoted security sources who have said the teenager appeared to be carrying a GoPro, a type of mobile camera that has been been used by terrorists to film their attacks.
"The man has asked for a negotiator and has threatened to kill a hostage if the security cordon is not moved back," according to the local France 3 TV.
Police told neighbours to stay indoors and set up a 300 metre security perimeter around the store.
A local businessman told newspaper La Depeche du Midi: “We have been asked to shut our shutters and to stay indoors.”
The national terrorism prosecutor's office said it was following the situation.
"It's worrying. It all happened so quickly," said Mr Texier, who expressed surprise at the incident in what he described as a quiet neighbourhood on the banks of the Garonne River.
He said the convenience store has been run for decades by a woman and her daughter who live in the adjacent building.
