Nazrul Islam died after falling from an eighth-floor area of a residential building on Siddheshwari Road in Ramna.
The incident occurred overnight between August 5 and 6 at the Rangs Garden Palace building.
Ramna police said officers went to Islam's flat to execute an arrest warrant issued in connection with a case in Gazipur.
Police described the warrant as linked to a CR, or complaint, case; one police account identified the case as involving alleged fraud.
Police said Islam was at or near an exterior ledge after becoming aware of their presence.
A Ramna police officer said investigators believed he may have been hiding on the ledge or trying to reach another flat when he fell.
The officer in charge of Ramna Police Station said officers had asked him to return inside before the fall.
Police recovered him injured and took him to Dhaka Medical College Hospital's emergency department.
A doctor there declared him dead in the early hours of August 6.
His body was kept at the hospital morgue for a post-mortem examination.
His age was reported inconsistently as 55 in three accounts and 51 in another.
Accounts also differ on whether he was a current or former Gazipur City Corporation accounts official.