The container in which 39 people were found dead was part of a convoy of three lorries, relatives of the suspected victims say.
They believe that over 100 migrants were being brought to the UK on the lorries.
Two of the vehicles are thought to have completed their trips but the third, which was carrying the deceased victims, was delayed at an unknown location.
The eventual destinations of the two other containers are not known.
Sky News cannot independently verify the accounts.
A priest in the town where many of the victims are thought to have come from also said he heard similar accounts from family members.
Father Anthony Dang Huu Nam, a Catholic priest from Vietnam’s Nghe An province, said he was aware of more than 100 people who were travelling to “a new life”.
“A few families confirmed the deaths of their relatives who are the victims of this tragic journey,” he said.
The priest earlier said he believed most of the victims were from neighbouring impoverished provinces in Vietnam.
Nam said he had been approached by a number of families who feared their relatives were among the dead.
Many victims of human trafficking are from Vietnam’s poorest regions.
Vietnam’s Prime Minister Nguyen Xuan Phuc has ordered an investigation into human trafficking allegations.
The bodies of 39 people were found in a refrigerated lorry trailer at Waterglade Industrial Park in Grays, Essex, on Wednesday.
DCI Martin Pasmore of Essex Police has appealed for the families of suspected victims to come forward.
He said police were investigating to establish “whether there is a wider conspiracy involved”.
Mr Pasmore told reporters he had spoken to the Vietnamese ambassador to the UK and added: “This is all about identifying our deceased, locate families and try to reunite them.”
The father of Nguyen Dihn Tu, who is suspected to have died, said: “He was on that truck. All 39 was dead and he must be one of them. He is gone now. There is nothing left of him. He died.”
Nguyen Dihn Sat told Reuters that relatives had called and told him his son was in the truck.
He said his son had been in the military and went abroad to seek work after being discharged.
Another victim is believed to be Anna Bui Thi Nhung from Nghe An province.
Her relatives said the 19-year-old had paid an agent over $10,000 (£7,915) in the hopes of travelling to the UK to become a nail technician.
“Many families in Yen Thanh have gotten rich from money sent back by their children working abroad,” said Le Dình Tuan, a neighbour who was visiting the family.
The father of Nguyen Dinh Luong, 20, also fears his son is among the dead.
He has been unable to get in contact since Luong said he would be joining a group in Paris to travel to the UK.
The family of a 26-year-old Vietnamese woman say they received harrowing messages in which she said she could not breathe and that she was “so sorry”.
Her father, Pham Van Thin, told CNN: “The smugglers said that this was a… safe route, that people would go by airplane, car… if I had known she would go by this route, I would not have let her go.”