MOSCOW (Reuters) - Russia's Federal Security Service (FSB) forces killed eight Islamic State militants on Sunday and six other Islamist rebels on Monday in the North Caucasus, the national Anti-Terrorist Committee (NAK) said.

Moscow is struggling to quell an insurgency by militants who have proclaimed a caliphate in the North Caucasus, a patchwork of mainly Muslim republics on Russia's southern rim, where separatists fought two wars in the 1990s.

NAK said the rebels killed on Sunday in the republic of Ingushetia were involved in "terrorist crimes" including killing law enforcement officials and extorting money.

On Monday, six insurgents suspected of involvement in "international terrorism" were killed in Nalchik, the capital of Kabardino-Balkaria, another mountainous North Caucasus republic, NAK said.

NAK identified one of those killed in Ingushetia as Adam Tagilov, who it said was behind fighting in the city of Grozny, capital of Chechnya, that killed more than 20 people - policemen and militants -- in December, 2014.

That fighting broke out hours before Russian President Vladimir Putin was due to give his annual state of the nation speech and put in question the authority of Ramzan Kadyrov, the Kremlin-appointed strong-arm head of Chechnya.

Rights activists and Kremlin critics accuse Russian security services of heavy-handed tactics in the North Caucasus, saying they target not only militants but also any dissenters.

(Writing by Gabriela Baczynska; Editing by Louise Ireland)