We propose a model to explain the background seismicity occurring at Somma-Vesuvius in its present, mostly quiescent period. A finite element procedure has been used to simulate the stress field due to gravitational body forces in an axisymmetric volcano characterized by a central high-rigidity anomaly. Results emphasize the important effect of axial high-rigidity, which concentrates at its borders stresses resulting from the gravitational load of the volcanic edifice, as well as external (regional) stresses. The joint effect of the gravitational loading and of the presence of the anomaly produces stresses very close to or above the critical rupture threshold. The observed spatial concentrations of seismicity and moment release correlate well with peaks of computed maximum shear stress. Seismicity is then interpreted as due to small stress perturbations concentrated around the high-rigidity core and added to a system already close to the failure threshold. This model can explain the widely observed occurrence of background seismicity at central volcanoes worldwide.