He was initially taken to Invercargill Hospital but was transferred to Dunedin Hospital yesterday afternoon.
Senior Sergeant James Ure said he remained in a critical condition there this morning.
Police said in a statement yesterday that they were still trying to ascertain exactly what had occurred, but it appeared to have been a "tragic incident".
"Police are currently working with the school administration, the Ministry of Education and other emergency services to ascertain what exactly has happened."
Children, parents and staff at the school were "clearly very upset".
"We have two of our traumatic incident staff at the school to provide support and we will make that support available for as long as it is needed," the statement said. "Our thoughts are with the injured boy and his family."
Police will continue working with the family, school and the Ministry of Education while staff conduct their enquiries.