"My thoughts are with all those who lost their lives & their loved ones."
Police said the truck was thought to have entered Britain at Holyhead, a North Wales port that is a major entry point for traffic from Ireland, on Saturday and to have originally started its journey in Bulgaria.
The victims, 38 adults and one teenager, were pronounced dead at the scene after they were found at the Waterglade Industrial Park, not far from docks on the River Thames.
Bulgaria's foreign ministry said it could not confirm at this stage whether the truck had started its journey from the country.
"We are still checking the information, published in the British media and we're contacting the authorities," foreign ministry spokeswoman Tsvetana Krasteva said.
Police officers in forensic suits were on Wednesday inspecting a large white container on a red truck next to warehouses at the site.
Police had sealed off the surrounding area of the industrial estate with large green barriers as they carried out their investigation.
"This is a tragic incident where a large number of people have lost their lives. We are in the process of identifying the victims, however, I anticipate that this could be a lengthy process," Essex Police Chief Superintendent Andrew Mariner said.
"We have arrested the lorry driver in connection with the incident who remains in police custody as our inquiries continue."
Home Secretary Priti Patel tweeted she was "shocked and saddened by this utterly tragic incident".
Dan Peters, from a hydraulic maintenance company on the industrial estate where the bodies were found, said it was "terrible" news.
He said the industrial estate was made up of courier and logistics companies.
In Britain's biggest illegal immigrant tragedy in 2000, British customs officials found the bodies of 58 Chinese people crammed into a tomato truck at the southern port of Dover.
