The 31-year-old, who was living at an asylum reception centre in the Norwegian town of Årdal, was due to be deported to Spain after having his application rejected in June, Norway's VG newspaper reported.
"This person had applied for asylum, and come to Norway in April," a spokesman for Norway's immigration directorate said.
"He was rejected in June, and was supposed to be returned to Spain under the so-called Dublin Regulation."
The Dublin Regulation is designed to stop people applying for asylum in multiple European Union member states. Under the system, the country where asylum applicants first arrive in the European Union is responsible for processing their applications.
The South Sudanese man is accused of killing the bus driver, Arve Haug Bagn (55) and Margaret Molland Sanden, a 19-year-old biotechnology student at the Oslo and Akerhus University College of Applied Sciences. The third victim, a Swedish man in his 50s, has not been named.