It has cited the ultra-hard-line Sunni Muslim group, which sees Shias as apostates, as saying "soldiers of the Caliphate in Bangladesh" detonated explosive devices in Dhaka during "polytheist rituals".
The series of blasts killed at least one person and wounded dozens as Shia Muslims gathered for a procession in the old part of Dhaka to mark the holy day of Ashura, police said.
However, Bangladesh's home minister told Reuters that no militants were involved and the blasts were not linked to an attack that killed 16 people at a Shia procession in neighbouring Pakistan hours earlier.
But soon after, threat monitoring group SITE reported that the IS had claimed responsibility for the bombings in Dhaka.
IS had also claimed responsibility for the recent murders of two foreign nationals in Bangladesh.
But the government and law-enforcing agencies ruled out any presence or activity of the militant group in the country.
Those injured in Saturday’s blasts were admitted at Dhaka Medical College Hospital (DMCH) and Sir Salimullah Medical College Mitford Hospital after the explosions.
Most of those at DMCH were in stable condition, an official there said.
BNP acting Secretary General Mirza Fakhrul Islam Alamgir demanded a "neutral investigation" into the attack.
"This is a clear sign of a deteriorating law and order," said acting Jamaat-e-Islami chief Moqbul Ahmed.
The government, however, has blamed the BNP and its ally Jamaat for the bombings.
US Ambassador Marcia Stephens Bloom Bernicat released a statement saying she was shocked by the attack and noted Bangladesh's long tradition of religious tolerance and communal harmony.