Three other members of the same family are also in intensive care after the incident.
Two members of a family from Cardiff have died and three more are in intensive care in Bangladesh.
Bangladeshi Police said the family were staying in a flat near the north-eastern city of Sylhet.
Police officers broke into the building on Tuesday morning after relatives raised the alarm.
It is understood father Rofikul Islam and his son Michael were found dead while his wife and two other children have also been taken to hospital.
The family, from the Riverside area of the city, was on a two-month visit to the country, local police have said.
Supt Farid Uddin said the condition of those in hospital had since shown some improvement.
He said all five family members were sleeping in the same room on Monday night and family members raised the alarm at about 10:00 local time when they failed to wake up.
At first the incident was thought to be food poisoning, but is now being investigated by police.
Post mortem examinations have been carried out and been sent to Dhaka. The results could take five days.