Guxim Imerio, 25, left the facility in South Lanarkshire about 9:40pm on Monday.
Police said he is not known to have family in the UK, but has links to Manchester and they want to “ensure that he is safe and well”.
He is 5ft 6’ with short black hair and was wearing a light-grey hooded top with a green Liverpool football top underneath when last seen.
Officers said he is not fluent in English.
A Police Scotland spokesman said: “Guxim has no knowledge of the local area and Police Scotland wish to ensure that he is safe and well.
“I would encourage any member of the public to contact police immediately if they have any information on the whereabouts of Guxim Imerio or have any information which could assist Police Scotland in tracing him.
“If you live in the vicinity of Dungavel, you may have seen someone of similar description in the local area walking, seeking shelter or hitchhiking?”