* Russian security forces fight gunmen in North Caucasus
* Moscow trying to root out militants before Winter Games (Adds details on explosives, law expanding FSB powers)
By Maria Kiselyova
MOSCOW, Jan 15 (Reuters) - Three members of the Russian security forces and four gunmen were killed in a shootout on Wednesday during a sweep for Islamist militants who have threatened to attack the Winter Olympics that begin in Sochi next month.
After two suicide bombings in southern Russia last month, Moscow is on high security alert. President Vladimir Putin has staked much personal and political prestige on the success of the Games, and put security forces on combat footing in Sochi.
Russia's National Anti-Terrorism Committee (NAC) said the dead militants included a man accused of carrying out a car bomb attack in the city of Pyatigorsk late last year which killed three people.
The shootout occurred on the same day that Russia's Duma, or lower house of parliament, introduced legislation aimed at broadening the powers of security services and boosting their oversight of the Internet, Russian news agencies said.
Voting has yet to be held on the bill, which the daily Vedomosti said was worked out during closed sessions held between parliamentarians and representatives of the Federal Security Service and the Interior Ministry.
The group of militants had been trapped in a house in the village of Karlanyurt in the Dagestan region of the North Caucasus, the NAC said in a statement. Five officers were also wounded in what a spokesman called a special operation.
Police defused two explosive devices at the house where the militants battled security forces. Television showed images of gray plumes of smoke rising later from a house in the impoverished North Caucasus region.
The Dagestan capital, Makhachkala, is about 620 km (385 miles) east of Sochi. The mostly Muslim region is plagued by bombings and shootings target state and police officials as part of the militants' fight to create an Islamist state there.
At least 34 people were killed last month in the suicide bombings in the southern city of Volgograd. Putin ordered safety measures to be beefed up nationwide after the attacks.
About 37,000 personnel are now in place to provide security in Sochi, located on the Black Sea and on the western edge of the Caucasus mountains. The International Olympic Committee has expressed confidence the Games will be safe.
But, underlining the danger of attacks, security forces said on Saturday they had arrested five members of a banned militant group in southern Russia and defused a homemade bomb packed with shrapnel.
The main spokesman for Russia's Investigative Committee, whose responsibilities include looking into bombings and other attacks, appealed to civilians on Tuesday to be more vigilant and help avert the threat of "terrorist" attacks.
The insurgency is driven by a mix of religious fundamentalism and anger among local residents over corruption and strong-arm efforts by appointed, pro-Moscow regional leaders to clamp down on militants. (Additional reporting by Thomas Grove, Editing by Timothy Heritag and Mark Heinrich)