The victims were found in a container on an industrial estate last week and were initially thought to be Chinese.
But Essex Police said it was now in "direct contact with a number of families in Vietnam and the UK" and the Vietnamese government.
A number of Vietnamese families have previously come forward fearing their loved ones are among the dead.
Pham Thi Tra My, 26, sent her family a message on Tuesday night, saying her "trip to a foreign land has failed".
Post-mortem examinations are being carried out on the 31 men and eight women to establish the cause of death.
Assistant Chief Constable Tim Smith said: "At this time, we believe the victims are Vietnamese nationals, and we are in contact with the Vietnamese Government."
He said police were not in a position to identify any of the victims.
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