The blaze erupted at about 1:30 am on Monday (2230 GMT Sunday) in the container in which the workers slept, the embassy's labour counsellor told AFP.
"They are burned so badly that it is very difficult to identify" the remains, but DNA analysis will be done, said the counsellor, asking not to be named.
The incident occurred at Naser Furniture Factory in the Gulf coast city of Dammam.
Police found five bodies in the container in which six people, including a Pakistani and an Indian, normally lived.
Saudi Arabia employs millions of foreign workers who perform a range of jobs from manual labour to management.