A 31-year-old Bangladeshi man, identified as Rubel, was injured near the Maheshpur border in Jhenaidah on the night of 3 August.
Munshi Imdadul, an assistant director of 58 Border Guard Bangladesh, said one person had been injured and was in Dhaka.
Accounts attributed to local residents and border-security sources said an Indian BSF patrol fired rubber bullets during the incident.
The incident was described as occurring in the Shyamkur border area of Maheshpur upazila.
A BGB account said several Bangladeshi youths had attempted to enter India to bring cattle.
Local residents said Rubel was with several people when they allegedly attempted to cross into India unlawfully.
The accounts differ on the asserted purpose of the attempted crossing, with only one specifying cattle.
He was first taken to a hospital in Jibannagar and then to Jashore Medical College Hospital before being moved to Dhaka Medical College Hospital.
The BGB official said the force had spoken with Rubel’s family.