In this study, the effects of non-uniform magnetic fields on melting and solidification of Nanoparticle-Enhanced Phase Change Materials (NEPCM) in an annulus enclosure are numerically investigated. Magnetic fields are applied on electrically conductive magnetic nanofluids by positioning a wire carrying the electric current in the center of the annulus. For the numerical simulation, a homogenous single-phase model and finite volume method are used and the melting and solidification processes are studied using the enthalpy-porosity method, where, instead of explicitly tracking the liquid-solid interface, the so-called liquid fraction quantity is computed based on the enthalpy balance in each cell and in each time iteration. The results show that, for the case with non-electrical conductive magnetic nanofluids, by increasing the magnetic number, the time required for the melting and solidification processes are reduced up to 39.91% and 14.29%, respectively. However, for the case with electrical conductive magnetic nanofluids at Ra = 10(4) and at specific magnetic numbers, the rate of both melting and solidification processes decreases by increasing the Hartmann number.