Nanocomposites made of magnetic spinel oxide Co1+yAl2-yO4 nanoparticles (NPs) with y = 0.93, 1.20 and 1.34 dispersed in an amorphous SiO2 matrix were prepared by a sol-gel method. The average crystallite size DXRD of about 25 nm determined by X-ray diffraction (XRD) agrees well with the average particle size observed in transmission electron microscopy (TEM). Dynamic magnetic susceptibility measurements indicate a spin-glass-like behavior of the nanoparticles below the spin-glass (SG) transition temperature TC < 10 K. Field dependent magnetization exhibits weak ferromagnetism and shift of the magnetic hysteresis loop. On the basis of the observed magnetic properties, we propose a model of the core-shell structure of the Co1+yAl2-yO4 nanoparticles with the spin-glass phase in the core and weak ferromagnetism in the shell. The exchange bias field disappears above the spin-glass transition temperature TC, whereas the weak ferromagnetism persists also at temperatures above TC. The component of weak ferromagnetism is associated with the presence of Co vacancies in the nanoparticle spinel crystal lattice. •Co1+yAl2-yO4 nanoparticles (NPs) exhibit spin-glass (SG) behavior below TC < 10 K.•Hysteresis loop recorded at T > TC reveals the existence of weak ferromagnetism (FM).•Weak FM is associated with the presence of Co vacancies in the NPs spinel lattice.•Horizontal shift of the hysteresis loop at T < TC is attributed to exchange bias effect.•A model of the NPs structure is proposed with SG in the core and weak FM in the shell.