Sprem, who suffered from multiple myeloma, died at the MD Anderson Cancer Center in Houston early Sunday morning, Central European Time. He was 56.
A commemoration will be held in parliament on Wednesday.
Sprem's body will lie in state at Zagreb's Mirogoj cemetery on Thursday morning. He will be buried at noon.
Culture Minister Andrea Zlatar Violic extended her deepest condolences and regret at Sprem's death, saying he gave in his public and political activity dignity to modesty, responsibility and equality.
She described him as a dear colleague, a committed politician who performed his duty as parliament speaker in a dignified way of which all Croatians could be proud.
Zlatar Violic also said Sprem's work, honesty and activity were inspiring, supportive and contributed to a feeling of security.