  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 $\simeq 5.7\times 10^3$ M$_{\odot}$. 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. 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
$L_X^{Obs}\simeq 2.7\times 10^{33}$ 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 $3\times 10^{-9}$. This measurement could be tightened
if moderately deep radio observations of the field were performed.
