Russian positions came under attack 14 times as the Chechen rebel's unilateral ceasefire expired last week, with one soldier killed in shootouts, the source said in Vladikavkaz on Monday.
Another soldier was killed and three injured when a Russian Ural truck came under attack on the outskirts of Grozny.
Four men died and four were injured in a battle near the southern Chechen town of Nozhai-Yurt. Russian officials said two separatists were killed.
And another soldier was killed and two were injured in a battle near the Chechen town of Itun-Kala, the source said.
Russian casualties are rarely reported on state television and the toll from the second Chechen war remains largely unknown, with the Russian government last year reporting that 4,500 soldiers were killed.
Human rights groups estimate that the true toll may be three times higher, since the Russian figure only takes into account soldiers killed on the battlefield, and not those who die from their wounds in hospital.