Police said Bacchu Chowdhury, 40, died on the way to hospital after the blast late on Wednesday near the shop he runs.
Chowdhury was a local leader of the Jatiya Party, the third-biggest party in the Bangladesh national parliament.
Nasir Ahmed, the local police chief, said the outlawed Maoist group Janajuddha (People's War) claimed responsibility for the killing in a telephone call to a local Bengali-language newspaper.
The murder follows a bomb attack in the city last month against a local leader of the main opposition Awami League party.
Sheikh Yunus Ali, an elected Khulna councillor, lost his right hand and suffered splinter wounds to his back after unidentified attackers threw two bombs at him.
The Maoist group has been linked to dozens of killings of politicians and journalists in the past decade in Khulna and other southwestern districts.
A bomb attack on Tuesday in nearby Jhenidah wounded four people. The attack, said by police to be possibly linked to a Maoist group, was the town's third in a week.