Explosions were heard at a warehouse in Dammartin-en-Goele, where two suspects in the Charlie Hebdo magazine shootings had been holding one hostage.
Explosions and gunfire could also be heard at a supermarket in eastern Paris, at Porte de Vincennes, where several hostages were held.
Reports suggest a gunman there was linked to the Charlie Hebdo suspects.
After the operation started, several hostages could be seen leaving the supermarket.
Police told French media that four hostages in the supermarket had been killed prior to security forces storming the site.
The hostage at the warehouse in Dammartin has been freed, while a police officer at the scene was injured, AFP news agency said.
Twelve people were shot dead and 11 were injured in Wednesday's attack on the office of Charlie Hebdo, a satirical magazine.
The unprecedented attack shocked France and there has been an outpouring of sympathy and solidarity worldwide.
The two suspects of the Charlie Hebdo shootings, brothers Cherif and Said Kouachi, had been on the run for two days, before being surrounded at the Dammartin warehouse on Friday.
French police said they came out firing, at which point police stormed the warehouse.