Porous spherical composites are widely found both in natural objects and in technical applications. To describe the mechanical behavior of such structures, it is necessary to take into account the interaction of solid and fluid phases, since the fluid inside the porous body supports and redistributes part of the external load. This paper proposes a model of a spherical composite including a porous core and an elastic transversely isotropic shell. The developed model is applied for stress and strain analysis in different loading regimes, including the effect of external normal pressure and the injection of an additional volume of fluid, for example, when modeling intravitreal injections in biomedical research. The analysis has shown that when modeling intravitreal injections and describing the eyeball as a poroelastic composite, the drop in intraocular pressure with increasing degree of anisotropy is less significant than in the case of a model with an elastic shell under the influence of internal pressure alone. The increase in the degree of anisotropy is more significant for the pressure reduction in the porous core in the mode of external normal pressure application than the injection of additional fluid volume. The study also investigates the influence of the geometry and mechanical properties of the porous core on the variation of the shell thickness. The results obtained provide a comprehensive understanding of the stress distribution and fluid pressure, which allows to consider the influence of shell anisotropy, core porosity and their mechanical characteristics on the behavior of spherical composites.