Rare earth elements (REEs) are strategically critical in the manufacture of advanced materials. Red mud and end-of-life NdFeB magnets can be good secondary sources for REEs, but recovery is difficult due to the high iron oxide content and small amount of REEs. Oxide mixtures whose composition of Fe, Nd, and Ga was similar to that in red mud were employed in experiments. In this study, a relatively inexpensive non-aqueous system was used to selectively dissolve Nd2O3 in a mixture with Fe2O3 and Ga2O3. The addition of ethylene glycol (EG) to HCl and H2SO4 solution depressed the dissolution of Fe2O3 and Ga2O3 from the mixtures, and thus selective dissolution of Nd2O3 was possible. The optimum conditions were as follows: (a) 1.0 M HCl in EG, 25 °C ± 1 °C, 50 g/L pulp density, 120 min, 200 rpm; and (b) 0.05 M H2SO4 in EG, 25 °C ± 1 °C, 50 g/L pulp density, 60 min, 300 rpm. Under these conditions, Nd2O3 was completely dissolved, whereas no Fe2O3 or Ga2O3 was dissolved by the H2SO4 system, and the dissolution percentage of these two oxides by the HCl system was less than 1%. Due to the selective dissolution of Nd2O3 from the oxide mixtures, it is simple to recover Nd. An efficient process can be developed for the recovery of REEs from red mud and end-of-life NdFeB magnets by applying our results.