Spherical implosion is the core process for achieving extreme compression in inertial confinement fusion (ICF) and weapons physics research. Its inherent geometric convergence effects make the evolution of interface instabilities significantly more complex than in planar cases. To address this physical problem, a potential flow model and elastic constitutive equations are introduced to derive and numerically solve the coupled equations governing the motion and stability of a spherical interface. Furthermore, the influences of material elasticity and external driving pressure on interface evolution and perturbation growth are investigated. Theoretical analysis indicates that the motion of the spherical interface is independent of the perturbed mode. The interface motion undergoes four stages: accelerated contraction, decelerated contraction, accelerated expansion, and decelerated expansion. Since the proposed model considers a purely elastic solid, the interface radius exhibits undamped periodic oscillations. As the shear modulus and driving pressure increase, the oscillation frequency of the interface rises. Specifically, elasticity suppresses the contraction depth of the interface, whereas the external pressure promotes it. With increasing driving pressure, the contraction ratio tends toward a constant value, suggesting that further increasing pressure has a limited effect on enhancing compression depth. The growth of the perturbation amplitude exhibits a characteristic of initially accelerating and then decelerating with the increase of the perturbation mode number until stabilization is reached. The perturbation growth function is introduced to quantify the evolution of the amplitude, showing that the extent of perturbation growth is positively correlated with the maximum value of this function. The results demonstrate that elasticity suppresses the development of instability, particularly inhibiting short-wavelength perturbations. For a specific perturbed mode, a cutoff shear modulus threshold exists, above which the perturbation is completely suppressed. Conversely, the driving pressure promotes interface instability and significantly enhances the growth of short-wavelength perturbations. Furthermore, the investigation within the parameter space of driving pressure and shear modulus reveals that increasing the perturbation mode number expands the stable parameter range, thereby raising the external driving pressure required to trigger interface instability for the same material. Finally, a comparison between the theoretical model and numerical simulations confirms the validity of the model in capturing the interface dynamic characteristics of complex media. This work advances the understanding of Rayleigh-Taylor instability in dynamic spherical geometries and offers a useful theoretical framework for further exploration of nonlinear behavior and elasto-plastic transitions, particularly in conjunction with numerical simulations.