The many-body potential for ferromagnetic and paramagnetic rhodium clusters proposed in this work has 11 parameters (14 for the paramagnetic case) that are fitted on the energy surface of Rh-2 through Rh-6 clusters calculated from first principles within the generalized gradient approximation (GGA) of density functional theory. Under this potential the most stable ferromagnetic and paramagnetic cluster structures are generated up to Rh-58. Additionally, the growth under several symmetries is pursued up to N=400. The face-centered-cubic (fcc) growth path is the most stable at that cluster size regime. An effective measure of the cluster stiffness is calculated as a function of cluster size displaying a monotone increase towards the bulk value. The melting temperature is about constant up to clusters with 45 atoms, presenting a sharp increase towards the bulk value at larger sizes.