News agencies on Tuesday reported that a suspected participant in last week's rebel attacks was killed after resisting police.
Local television urged residents not to leave their homes if possible, and local schools advised parents to take their children home.
Shooting was heard in the suburb of Dubki, where the city's main mortuary is and where soldiers were conducting a sweep for suspected rebels, said Luiza Orazayeva, an activist with the Memorial rights organisation. An Associated Press reporter heard gunfire on the southwestern edge of the town and in another district, near a police precinct.
Police cordoned off streets blocks away from the area and allowed neither cars nor pedestrians to enter.
Armoured personnel carriers were parked in the streets, and there was a strong smell of smoke, though the source was unclear.
Officials sources said fighters conducted a coordinated series of attacks on police and other government buildings in Nalchik on Thursday, and 137 people were killed.
Shamil Basayev said he was
behind the Nalchik attack (file)
Russian news agencies reported that police killed a man early on Tuesday when he allegedly put up resistance during a document check.
The ITAR-Tass news agency quoted an unnamed source in the regional Interior Ministry as saying the suspect was killed after he refused police demands to stop and tried to take a machine gun from under his jacket.
The suspect had taken part in last week's attacks, the Interfax news agency quoted Interior Ministry spokeswoman Marina Kyasova as saying. She said he had spent the past few days in a forest outside the town and tried to sneak home overnight.
Two other men who resisted managed to escape, Interfax reported, but ITAR-Tass quoted the regional Interior Ministry as saying only one man was involved in the confrontation.
Chechen regional commander Shamil Basayev, the purported author of recent attacks in Russia, claimed he was behind last week's attacks in Nalchik, according to a statement posted on a Chechen rebel-connected website.
Basayev said the attacks were carried out by fighters affiliated to the Chechen separatists, but that Chechen fighters were not involved.