Nanayakkara claimed that the troops had captured five positions close to a major LTTE base in the north.
The clashes were in the Mannar, Vavuniya, Weli Oya and Jaffna regions.
The LTTE did not immediately comment on the events.
Independent confirmation of the deaths could not be gained as access to the regions is heavily restricted.
Fighting has escalated since the start of the year when the government withdrew from a Norwegian-brokered truce with the LTTE.
The government is aiming to eliminate LTTE control of northern territory on the island.
The LTTE has been fighting for a separate homeland for ethnic Tamils since 1983. More than 70,000 people have been killed in the civil war.