The rare earth elements (REE) include the group of 15 lanthanides, scandium and yttrium and have diverse applications in technological and nuclear areas. The existence of REE in massive solid mining wastes generated in leaching processes of copper minerals in the Atacama region of Chile generates the possibility of creating added value to the treatment of this type of waste and supporting the development of a circular economy, generating a useful by-product in different industries. In order to know the behavior of these elements present in the solid carrier waste, a leaching process was carried out by using two agents separately, corresponding to hydrochloric and nitric acid. The technical feasibility to recover REE from carrier tail was demonstrated, the best leaching agent for these elements being a hydrochloric solution, obtaining a maximum recovery efficiency of 64.5%, for an acid concentration: 3M, temperature: 40°C and (liquid/solid) ratio: 4. Lanthanum and cerium present the best individual recoveries compared to the other REE, with a maximum efficiency for a hydrochloric solution of 75.7% and 70.0%, respectively. The interaction of operational parameters that most influence the REE recovery corresponds to the temperature and the (liquid/solid) ratio. After 4 hours of leaching, REE recovery efficiencies remain practically constant. Acid consumptions correspond to 11 (kg HCl/ton mining tail) and 29 (kg HNO3/ton mining tail). The highest amount recovery ratios of these elements correspond to 0.355 and 0.224 (kg REE/ton mining tail), for hydrochloric and nitric solutions, respectively. These results influence the types of reagents and parameters to be studied in the following stages of the global process.