The driver, a 25-year-old man from Northern Ireland, was arrested on suspicion of murder, the Efe news reported.
A public official in Northern Ireland identified the driver as Morris "Mo" Robinson of Portadown in County Armagh.
Authorities in the county of Essex, east of London, were called out to the Waterglade Industrial Park in Grays before 1.40 a.m. on Wednesday.
The 38 adults and one teenager found in the trailer were pronounced dead at the scene.
Police said that the container arrived in Purfleet, Essex, just after 12.30 a.m., and was hitched to a tractor unit that originated in Northern Ireland.
The rig left the port before 1.10 a.m. and ambulance crews reported the discovery of the bodies about 30 minutes later.
The container appeared to be a refrigerated unit, according to Richard Burnett, the director of the UK Road Haulage Association, who said that the conditions inside would have been "absolutely horrendous," with temperatures as low as minus 25 C (minus 13 F).
Deputy Chief Constable Pippa Mills said the "No. 1 priority" was to learn the names and origins of those who died.
"We are in the process of identifying the victims, however, I anticipate that this could be a lengthy process," she said.
Police in both Northern Ireland and the Republic of Ireland said they stood ready to support Essex authorities in the probe.
Addressing the Irish parliament, Prime Minister Leo Varadkar said his government would "carry out any investigations that are necessary if it's established the truck passed through Ireland."
The Belgian federal prosecutor's office also opened an investigation, a spokesman said.
The Bulgarian government confirmed reports from British police that the trailer was registered in that Balkan nation.
The registration for the trailer was in the name of an Irish-owned company, the Bulgarian foreign ministry said, while Prime Minister Boyko Borisov said the vehicle had not been in Bulgaria since 2017.
UK Prime Minister Boris Johnson described the deaths as an "unimaginable tragedy and truly heartbreaking."
"I know that the thoughts and prayers of all members will be with those who lost their lives and their loved ones," he said in Parliament.
In 2000, the bodies of 58 Chinese migrants were found inside a container in the UK port of Dover.
( With inputs from IANS )