The 30-year-old man, of no fixed address, was found in Birmingham at about 23:30 GMT on 29 November.
He was found in a loading area behind The Victoria pub, in John Bright Street, on what was one of the coldest nights of the year.
West Midlands Police said CCTV had been examined and an investigation was taking place.
A statement said: "Following examination of CCTV it is believed that as the man became unwell, someone searched his pockets and property may have been stolen."
The death has been referred to the coroner.
Police are investigating whether property was stolen from a man whose body was found in a city centre.