Šprem started his professional career with RK Zagreb in the late 1990s. He also spent a season with RK Medveščak Zagreb in 2001 and 2002, before returning to RK Zagreb for a second spell with the club between 2002 and 2004. He eventually left RK Zagreb for German side SG Flensburg-Handewitt in October 2004. He left Flensburg for a short spell with TuS Nettelstedt in February 2005, returning to the club in the summer of the same year and eventually spending a season with them. In the summer of 2006, he left Flensburg for a short spell with MT Melsungen, before joining his current club, HSG Nordhorn, in November 2006. He left Nordhorn in 2009.