Emergency services were called to the scene just after midnight but the fire was still burning on Wednesday morning, despite nine fire engines working to extinguish the huge blaze overnight.
A spokesperson for the region's police force told the TT news agency that his team had managed to get it under control, but explained that the entire building had been destroyed by the smoke and flames.
"The information we have received is that it started in a garbage bin next to the outside," said duty officer Magnus Lefèvre.
No one is understood to have been injured in the fire, but several cars had to be moved away from the area using a pick up truck as the flames spread.
A tweet from a regional newspaper showing the fire
"When we arrived, it was so impossible to extinguish, the focus was to limit it spreading and let it burn," Marco de Pedis, the leading firefighter at the scene, told Swedish tabloid Aftonbladet.
A witness takes a photo of the fire on Wednesday. Photo: Frida Winter/TT
Neighbours were warned to stay inside and close their doors and windows to avoid the possibility of smoke inhalation.
However de Pedis said that the heavy smoke should fade from the area in the coming hours.
Trelleborg is the southernmost town in Sweden and has a population of around 43,000. It is situated around 30 kilometres from Malmö, Sweden's third largest city and is a major hub for asylum seekers, many of whom arrive by ferry from Germany.