The Algerian army says it has killed a man suspected of kidnapping and beheading a French hiker, following France's decision to launch air strikes against the Islamic State in Iraq and the Levant (ISIL) in Iraq.

Tayeb Louh, Algeria's justice minister, said on Wednesday that the army killed a man last month who was suspected of taking part in the beheading of Herve Gourdel.

"One of the suspects was eliminated in October by the army during an anti-terrorist operation," Louh told reporters, citing a probe into the murder of the 55-year-old.

Gourdel was kidnapped while hiking in a national park in Algeria's Djurdjura mountains, once a magnet for tourists but later becoming a sanctuary for armed groups.

He was later beheaded in a video posted online by a group calling itself the "Soldiers of the Caliphate" after France rejected a 24-hour ultimatum to halt its air raids against ISIL.

The Algerian army combed the mountains where the September 24 beheading took place, allegedly killing several fighters and finding the campsite where the video was shot.

The "Soldiers of the Caliphate" split from al-Qaeda in the Islamic Maghreb in September and pledged allegiance to ISIL and its leader Abu Bakr al-Baghdadi, who has declared himself the Caliph of all Muslims.