The incident came two days after Indian border troops were accused of killing two other farmers across the frontier.
Bangladeshi farmer Uttam Kumar was working in his rice field on Monday when Indian border guards fatally shot him in Lalmonirhat district, 260km north of the capital, Dhaka, an officer from the Bangladesh Rifles - a paramilitary border force - said on Monday on condition of anonymity.
Indian guards gunned down another farmer, Ashraful Islam, also on Monday, in nearby Thakurgaon district, the officer added, citing witness accounts. The Indian guards carried both the bodies into Indian territory, according to him.
India said its Border Security Force (BSF) was not involved in the shooting. ''We deny these accusations,'' Kamal Kumar, a BSF spokesman in New Delhi, said.
Tensions have been high along the frontier since an Indian guard and a 10-year-old Bangladeshi girl died in cross-border shooting between Indian and Bangladeshi troops on 16 April.
Each side accused the other of provoking the violence.
Indian border guards have been
blamed for a number of deaths
The 4000km-long border is unclear in places. Skirmishes between villagers or guards often take place along the frontier, where smuggling and human trafficking are common.
On Saturday, Bangladesh accused Indian border guards of fatally shooting two Bangladeshi farmers and taking their bodies to a morgue in India, returning them on Monday following repeated Bangladeshi requests.
An Indian border officer initially denied the attack took place, but local authorities later acknowledged the presence of the bodies.
The Bangladesh Rifles asked India to return the bodies of those allegedly killed on Monday. ''We have an agreement not to shoot straight at innocent villagers, even if they stray close to the border,'' the Bangladeshi officer said.
The Bangladesh Rifles officer had earlier said that Indian troops on Sunday fired at a dozen children as they picked mangoes in Bangladesh, but no one was hurt.