Bui Thanh Son, Vice Minister of Foreign Affairs, said Monday that Vietnam's Ministry of Public Security has received the files for "connecting the information and verify the victims' idenfication."
It would be difficult to hasten the investigation as verifying the 39 identities would require DNA test results, said Son.
"As of right now, it could not be ascertained whether there were Vietnamese among those who died inside the container," he said.
Authorities of Nghe An Province in central Vietnam have taken hair and nail samples from families who reported missing members feared to be among the deceased 39 for verification purposes, said Nguyen Dac Vinh, the province's party secretary.
People should wait to hear official confirmation from authorities regarding the case, said Vinh.
As of Sunday, 24 families in Nghe An and its neighbor Ha Tinh Province have reported missing members in the U.K. and Europe.
Vietnam Foreign Ministry representatives would meet the British Ambassador in Vietnam on Monday for further information exchange regarding the case, said Son.
The Ministry of Public Security is also investigating possible involvement of individuals or groups who help Vietnamese emigrate illegally.
Prime Minister Nguyen Xuan Phuc on Saturday ordered a probe into the trafficking of Vietnamese citizens in the wake of the 39 deaths.
On October 23, U.K. emergency services discovered the bodies of 38 adults and one teenager, suspected immigrants, after being alerted of people in a refrigerated container truck at Waterglade Industrial Park in Grays, Essex County, east of London.
Police in Britain had charged Maurice Robinson, the 25-year-old truck driver, of Craigavon in Northern Ireland, with 39 counts of manslaughter and other offences including conspiracy to traffic people.
Three other people arrested in connection with the investigation had been released on bail until November 11-13.