Five extremist attacks have been thwarted in France since a January assault by Islamic extremist gunmen left 20 people dead, including the attackers, the French prime minister has said.

Manuel Valls said the five included one planned attack uncovered on Sunday when the would-be gunman apparently shot himself in the leg by accident and called for an ambulance, raising suspicions of the police.

"The threat has never been so significant," Mr Valls said as he pressed for passage of a measure that would give the government increased surveillance powers and give wider leeway for prosecutors.

Authorities say the 24-year-old Algerian arrested in Paris on Sunday was targeting one or more churches. The computer science student had been flagged as a security risk this year and in 2014, but managed to secure an arsenal of weapons.

He is also linked to the death of young Frenchwoman Aurelie Chatelain, whose bullet-riddled body was discovered on Sunday morning inside her burning car in the same Paris suburb as the targeted church.

"The threat was there, the plan was to act, there was a death - Aurelie Chatelain - and we cannot keep that a secret. We have to be lucid. We have to look reality in the face," Mr Valls said.

He did not detail the other planned attacks or how they were thwarted.