An attack last week also targeted
Two pro-Russian policemen were killed and another nine wounded, some of them seriously, when a bomb went off near the village of Dyshne-Vedeno, in the southeastern Chechen mountains, reported the Interfax-AVN news agency.
The area is now predominantly under separatist control.
The other Russian casualties came in shootouts in various regions of the shelled-out Chechen capital Grozny, said the news report.
The reports gave no casualty figure for the separatist fighters.
Last week at least 11 people, including nine children, were killed in an explosion that ripped through an apartment block in the Chechen capital. The cause of the blast was unclear.
The explosion came a day after a woman suicide bomber detonated explosives near a bus carrying Russian servicemen near the Chechen border, killing 19 people. Chechen officials denied involvement in the attack.
Separatists have continued waging their battle in Chechnya, despite a parliament decision last week approving amnesty for Chechen fighters.
The amnesty has outraged human rights groups since it also protects Russian soldiers accused of numerous atrocities against civilians in the republic.
The latest war between separatists and federal troops in the Caucasus republic broke out in October 1999, when Chechen fighters were accused of masterminding a series of blasts against residential Moscow apartment blocks.
Thousands of separatists, soldiers and civilians, have been killed in the fighting for an independent Chechen state.
Chechnya enjoyed de facto independence between 1996 and 1999 after rebels defeated Russian troops in an earlier separatist conflict.