Bombs exploded near an Eid prayer gathering in Sholakia in Kishoreganj district where at least 200,000 people had gathered, police said.
Police said one constable was dead and at least six others were injured.
One suspected attacker was also reported to have been killed in an exchange of fire with police at the blast site as roads in the area were cordoned off.
The incident comes after last week's deadly attacks in Dhaka which killed over 20 people mainly foreigners including an Indian girl.