In this paper, we investigated the effect of shape and size of core on the plasmonic response of spherical ZnO@Au core-shell nanostructures embedded in a passive host matrices of MgF2 within the framework of the qausistatic approximation. The absorption cross-section and local field enhancement factor of spherical ZnO@Au core-shell nanostructures are effectively studied by optimizing the parameters for a fixed composite diameter of 20 nm. In this two-layered core-shell nanostructures, four plasmonic resonances are found; the first two resonances associated with ZnO/Au and Au/MgF2 interfaces, whereas the third and fourth resonances are associated with the transverse and longitudinal modes, respectively. The peaks position and intensity of these resonances are varied by optimizing the shape and size of the core material. The tunability of the plasmon resonances of the composite systems enables it to exhibit very interesting material properties in a variety of applications extending from the visible to infrared spectral regions.