Security forces launched the operation in Macchil sector in north Kashmir's Kupwara district following information about presence of militants in the area, an Army official said on Monday.
One terrorist has also been gunned down in the ongoing encounter.
One militant and two BSF jawans were killed while five other security force personnel were injured in the initial firefight between the two sides, the official said.
Among the injured personnel -- three BSF jawans and two Army soldiers -- condition of one BSF jawan was critical.
The injured BSF jawan later succumbed to injuries, a police official said. He said the slain BSF personnel were identified as sub Inspector Mohinder Yadav, Head Constable C P Singh and Constable Babu Shaan. The operation was in progress when reports last came in, he added.
FLASH: Two BSF soldiers lost their lives, and one terrorist gunned down in an encounter in Machhil (J&K). Ops underway.— ANI (@ANI_news) August 8, 2016
(With inputs from PTI)