We employ numerical simulations to calculate the shape of the interface separating a nonmagnetic fluid droplet surrounded by a ferrofluid confined in a Hele-Shaw cell. The system is subjected to an azimuthal magnetic field generated by a current-carrying wire passing through the center of the nonmagnetic droplet, perpendicularly to the Hele-Shaw plane. The azimuthal field has a destabilizing effect on the interface and pulls the ferrofluid radially inwards. On the other hand, surface tension tends to stabilize interface deformations. The resulting dynamic behavior and morphological shapes are also influenced by the viscosities of both fluids and the magnetic susceptibility of the ferrofluid. In the current work, we employ an accurate boundary integral method based on the vortex-sheet formalism to examine the nonlinear dynamics and determine these interfacial deformations. We analyze how the interaction between magnetic, surface tension and viscosity effects affects the interface morphology. This is captured by varying the dimensionless parameters that control the flow. They are the magnetic susceptibility of the magnetic material, the viscosity contrast between the non-magnetic droplet and the ferrofluid, and the effective surface tension, which balances the effects of surface tension with the intensity of the applied field. We compare particular cases from our nonlinear simulations with theoretical predictions from a modecoupling perturbative analysis that retains linear and weakly nonlinear terms. The correspondence between the two approaches for short times corroborates the validation of the numerical method and indicates the validity and some limitations of the approximate approach.