Mir Sanowar Rahman, 55, was killed on Friday by machete-wielding attackers, who came on a motorcycle.
The attack also left his friend Md Saifizzuaman, teacher of Kushtia’s Islamic University injured. He is being treated at a Dhaka hospital.
US-based monitoring group SITE tweeted the claim citing IS’ Amaq News Agency a day after the attack, Reuters reported.
The doctor and his friend were riding a motorcycle on their way to a weekly free clinic he had been operating.
Police said the two long-time friends were fond of Baul ideology.
The attackers used their motorcycle to block Sanowar's motorcycle, before slashing him and his friend with machetes.
The murder followed a now-familiar pattern witnessed in the killings of teachers, bloggers and online activists as well as foreigners — a use of sharp weapons to target the head or neck of the victims.
IS have claimed responsibility for some attacks in Bangladesh, including the killings of two foreigners last year, and a Rajshahi University professor and a Hindu man in Tangail last month.
Al Qaeda in the Indian Subcontinent has also claimed some of the attacks, including the murder of gay rights activist Xulhaz Mannan and his friend in Dhaka.
The Bangladesh government, however, denies the presence of IS or al Qaeda in the country and says ‘home-grown’ Islamist militants were responsible for the killings.