Three militants stabbed Hossain Ali Sarkar, 68, to death while he was on his morning walk in the northern district of Kurigram.
Bangladeshi police say Mr Ali Sarkar converted from Islam to Christianity in 1999, however he was not a high-profile preacher.
At the time of his death last Tuesday they were unsure whether Islamist militants had murdered him, or whether the murder was linked to family disputes.
SITE Intel Group said on Twitter that Islamic State were behind the attack:
#ISIS claimed credit for the hacking death of Hussein Ali Sarkar, a Christian convert, in Kurigram #Bangladesh pic.twitter.com/5R7VjpgND1
— SITE Intel Group (@siteintelgroup) March 22, 2016
Tobarak Ullah, the Kurigram district police chief, said at the time of Hossain Ali Sarkar's murder: "They left the scene exploding crude bombs to create panic.
"We have started an investigation into the murder.
"So far we have not found any militant link."
Mr Ullah also confirmed five people had been arrested for questioning.
