Essex police says its aware that there may be people in the UK illegally who are concerned for a relative but are fearful of approaching the police but promised that “Essex Police will take no action whatsoever against an individual that comes forward,” he said.
“We will do what we can to identify them to see if they have a family member as a victim and to establish the reconciliation of that person and if necessary repatriation.”
The BBC says it has also been contacted by Vietnamese families who fear their relatives were among the eight women and 31 men found at an industrial park in Grays on Wednesday.
They were inside the refrigerated unit which had travelled to the UK from the Belgian port of Zeebrugge.
Essex Police initially said the victims were believed to be Chinese nationals.
VietHome said it began to receive reports of missing people soon after the discovery. Those who have disappeared were aged between 15 and 45, it said.
One woman’s brother told the BBC she had not been heard from since Tuesday, when she sent panicked texts saying she could not breathe.
The last message from Pham Thi Tra My, 26, was sent to her family at 22:30 BST on Tuesday – two hours before the trailer arrived at the Purfleet terminal from Zeebrugge in Belgium.
Her family have shared texts she sent to her parents which when translated, reads: “I am really, really sorry, Mum and Dad, my trip to a foreign land has failed.
“I am dying, I can’t breathe. I love you very much Mum and Dad. I am sorry, Mother.”
“We are asking the British police to help investigate so that my sister can be returned to the family,” her brother said.
Relatives of Nguyen Dinh Luong, 20, have also said they fear he is among the 39 victims.
His father, Nguyen Dinh Gia, said he had not heard from his son since last week when he said he was joining a group in Paris to try and reach the UK.
“He often called home but I haven’t been able to reach him since the last time we talked last week,” he said.
Mr Luong’s brother, Nguyen Dinh Oanh, wanted to warn others against making similar dangerous journeys.
He told the BBC’s Vietnamese service: “Those who still would like to go abroad, please think carefully, the chance for survival is 50-50.
“My advice to them is stay at home, work hard enough just to make a reasonable living. “We want the money but we value the human life more.”
All 39 bodies have been recovered from the trailer and work is continuing to identify those who died.
Three people were arrested on Friday on suspicion of manslaughter and conspiracy to traffic people.
A 48-year-old man from Northern Ireland was detained at Stansted Airport and a couple from Warrington, Joanna and Thomas Maher, were arrested in Cheshire.
Before their arrest, the couple denied having anything to do with the lorry and told reporters they sold it last year.
The driver of the lorry, Mo Robinson, has been held on suspicion of murder since Wednesday
Police have been given extra time to question the 25-year-old of County Armagh.
The trailer arrived in Purfleet on the River Thames from Zeebrugge in Belgium at 00:30 BST on Wednesday.
It left the port shortly after 01:05 the same day and the bodies were discovered at Waterglade Industrial Park about 35 minutes later.
Police in Belgium are trying to trace the route of the trailer and the driver who delivered it to Zeebrugge.
“We would like people to be arrested as soon as possible,” said a spokesman for the Belgian prosecutor’s office.
Bernie Gravett, a former Met officer who now advises the EU on trafficking, said identifying the victims would be a difficult process as those in the container may have been carrying false documents.
“It’s a cruel stage for the families, because hundreds if not thousands are currently on those routes, so I appreciate we are getting calls from Vietnam saying my loved one is missing and my loved one may be on that lorry but they could be on another lorry,” he said.
The Vietnamese Embassy in London has been in contact with Essex police since Thursday, a spokesman confirmed.
Liu Xiaoming, the Chinese Ambassador to the UK, said China was taking “firm and effective measures against human trafficking”.
He called on the international community to “step up co-operation to crackdown on the crime of human trafficking”.