Kajal Mahmud, a local leader of the governing Bangladesh Nationalist Party, was killed instantly in the attack late on Sunday in Meherpur district, said Awlad Husain, the local police chief.
The official said on Monday that Mahmud and three friends were having tea in Babundi village market when about four masked men on two motorbikes threw the bombs at them and then fled.
The friends were injured and were being treated in a hospital in Meherpur town, 175km west of national capital Dhaka, Husain said.
There was no immediate claim of responsibility for the attack. No one has been arrested.
Meherpur is a farming region, where Maoist fighters operate. Police say that the rebels attack wealthy people who refuse to pay them extortion money.
Husain said police were also investigating a possible link to the banned Islamic group Jamatul Mujahidin Bangladesh.
The organisation wants to establish Islamic law in Bangladesh, a predominantly Muslim nation governed by secular laws.