Deceased Sabil Uddin, 36, was the son of Abul Hossain, a resident of Pakhiura area under Narayanpur union in the upazila.
Locals said Sabil entered Indian territory through the border.
The BSF members opened fire on him while he was returning at about 2:00 am, leaving him dead on the spot.
Kochakata Police Station OC Mamun-or-Rashid confirmed the death.
Meanwhile, BGB-22 Battalion commanding officer Lt Col Jamal Hossain said it is yet to be ascertained whether it was a border killing or not.
The reason behind the death will be known once the autopsy is done, he added.
