Eleven of the victims were taken by an ambulance on Thursday night from the Port of Tilbury to Broomfield Hospital, Chelmsford, reports the BBC.
Post-mortem examinations will be the next step in the investigation.
The police have been granted extra time to question lorry driver Mo Robinson, 25, on suspicion of murdering the eight women and 31 men.
Essex Police announced on Thursday that the victims were all "believed to be Chinese nationals".
China's Ambassador to the UK Liu Xiaoming later tweeted: "The Chinese Embassy has sent a team led by the minister-counsellor in charge of consular affairs to Essex, England.
"They have met with the local police, who said that they are verifying the identity of the 39 deceased, whose nationality still cannot be confirmed."
Meanwhile, three properties in Northern Ireland have been raided and the National Crime Agency was working to establish if "organised crime groups" were involved, the BBC reported.
Police believe that the tractor unit - the front part of the lorry - had entered the country via Holyhead in Wales on Sunday, having travelled from Dublin.
The trailer arrived in Purfleet on the River Thames from Zeebrugge in Belgium at 12.30 a.m. on Wednesday.
The lorry and trailer left the port at Purfleet shortly after 1.05 a.m. the same day.
Ambulance staff discovered the bodies in the container at Waterglade Industrial Park in Grays at about 1.30 a.m.
Vigils were held outside the Home Office in London and at the front of City Hall in Belfast on Thursday.
( With inputs from IANS )