Essex Police announced on Friday evening that it believed it knew the nationality of all the victims.
Assistant Chief Constable Tim Smith said: “We are in direct contact with a number of families in Vietnam and the UK, and we believe we have identified families for some of the victims whose journey ended in tragedy on our shores.”
He said the force was awaiting “confirmatory evidence” in order to present the case to the coroner.
Mr Smith added: “This evidence is being gathered across a number of jurisdictions worldwide. As a result, we cannot at this time announce the identity of any of the victims.”
Le Thi Thu Hang, a spokeswoman for Vietnam’s Ministry of Foreign Affairs, said in a statement on Saturday: “This is a serious humanitarian tragedy. We are deeply saddened by the incident and wish to extend our heartfelt condolences to the victims’ families.
“We wish they may soon overcome those tremendous pain and losses.”
She said the Vietnamese embassy in the UK had been working closely with the British authorities to support families and help find the identities of the victims.
She said the British police and the Vietnamese Ministry of Public Security would publish their findings “as soon as possible”.
Ms Hang continued: “Vietnam strongly condemns human trafficking and considers it a serious crime subject to strict penalties.
“Vietnam calls upon countries in the region and around the world to step up cooperation in combating human trafficking in order to prevent the recurrence of such tragedy.”
She added: “We are grateful that the British Government and other relevant authorities have actively conducted the investigation, timely provided identification information to and worked in close coordination with the Vietnamese competent authorities.
“We hope that the British side [will] soon complete the investigation to bring those responsible for this tragedy to justice.”
The bodies of eight women and 31 men were found in a refrigerated trailer attached to a lorry in an industrial park in Grays, Essex, in the early hours of October 23.
The update from Essex Police regarding the victims’ nationalities comes after the force started extradition proceedings to bring 22-year-old Eamonn Harrison from Ireland to the UK.
Harrison, of Newry in Co Down, Northern Ireland, appeared in Dublin High Court on Friday charged with 39 counts of manslaughter, along with human trafficking and immigration offences, and was remanded in custody.
Detectives have also urged Ronan Hughes, 40, and his brother Christopher, 34, said to have links with the road haulage and shipping industries, to hand themselves in.
The pair, from Armagh in Northern Ireland, are wanted on suspicion of manslaughter and human trafficking.
It is understood Ronan Hughes spoke to police shortly after the driver of the lorry, Mo Robinson, 25, was arrested following the discovery.
Robinson, of Craigavon, Northern Ireland, appeared at Chelmsford Magistrates’ Court on Monday charged with 39 counts of manslaughter, conspiracy to traffic people, conspiracy to assist unlawful immigration and money laundering.
Three other people arrested in connection with the incident - two men aged 38 and 46 and a 38-year-old woman - have been released on bail.
In Vietnam, local media reported that police in the Ha Tinh province had arrested two people in connection with the deaths after launching an investigation into suspected human trafficking of workers.
The investigation was prompted when 10 families from Nghen Town, Thien Loc, Vinh Loc and Thanh Loc contacted authorities to say they had lost contact with their relatives.
It is still not known how the victims, who police initially believed were Chinese nationals, entered the lorry trailer before it travelled from Zeebrugge in Belgium to Purfleet in Essex.