A new class of high-entropy spinel oxides, (Cr0.2Fe0.2Mn0.2Ni0.2Zn0.2)3O4 and (Cr0.2Fe0.2Mn0.2Co0.2Zn0.2)3O4 nanocrystalline powders, has been synthesized by solution combustion synthesis. Their microstructures and magnetic properties were compared with those of (Cr0.2Fe0.2Mn0.2Co0.2Ni0.2)3O4. All the oxides were identified to have the spinel crystal structure (Fd-3m) with average crystallite size of about 24 nm, and their lattice parameters found to decrease with reduction in the electronegativity of Ni2+, Co2+ and Zn2+. Furthermore, nonmagnetic Zn2+ substitution for either magnetic Co2+ or Ni2+ weakened the ferromagnetic ordering and magnetic moments, showing the controllable magnetic property of this new class of high-entropy oxides.