Context. Although the Orion Nebula Cluster is one of the most studied clusters in the solar neighborhood, the evolution of the very low-mass members (M* < 0.25 M⊙) has not been fully addressed due to their faintness.Aims. Our goal is to verify if some young and very low-mass objects in the Orion Nebula Cluster show evidence of ongoing accretion using broadband VLT/X-shooter spectra.Methods. For each target, we determined the corresponding stellar parameters, veiling, observed Balmer jump, and accretion rates. Additionally, we searched for the existence of circumstellar disks through available on-line photometry.Results. We detected accretion activity in three young stellar objects in the Orion Nebula Cluster, two of them being in the very low-mass range. We also detected the presence of young transition disks with ages between 1 and 3.5 Myr.Key words: stars: low-mass / stars: pre-main sequence / open clusters and associations: individual: Orion Nebula Cluster