We investigate the existence of self-dual solitons with internal structure in a gauged O(3) nonlinear sigma model immersed in a dielectric medium generated by a real scalar field (dubbed the source field). We consider rotationally symmetric configurations and applying the Bogomol'nyi-Prasad-Sommerfield formalism to obtain the energy lower bound and the respective first-order differential equations (or self-dual equations). By solving such a system of equations for three different dielectric media, we find the internal structure generates relevant changes in the soliton profiles when compared with the ones obtained without the presence of the dielectric medium.