Context. By studying the optical brightness surface density of the globular cluster NGC 6388, it has been recently proposed that it harbors a central intermediate-mass black hole with mass ≃5.7 x 10 3 M ⊙ . Aims. We expect that the compact object in the center of NGC 6388 emits radiation in the X-ray band as a consequence of the accretion from the surrounding matter. We searched for XMM-Newton and Chandra observations towards NGC 6388 to test this hypothesis. Methods. We determine both the hardness ratios and luminosity with a minimum set of assumptions for each of the identified field sources. Results. The Chandra satellite disentangles several point-like X-ray sources, probably low mass X-ray binaries, well within the core radius of the globular cluster. However, three of them, coinciding with the cluster center of gravity, remain unresolved. Their total luminosity is L Obs X ≃ 2.7 x 10 33 erg s -1 . If one of these sources is the X-ray counterpart of the intermediate-mass black hole in NGC 6388, the corresponding upper limit on the accretion efficiency, with respect to the Eddington luminosity, is 3 x 10 -9 . This measurement could be tightened if moderately deep radio observations of the field were performed.