We present a study of X-ray active galactic nucleus (AGN) overdensities in 16 Abell clusters, within the redshift range 0.073 < z < 0.279, in order to investigate the effect of the hot inter-cluster environment on the triggering of the AGN phenomenon. The X-ray AGN overdensities, with respect to the field expectations, were estimated for sources with Lx ≥ 1042 erg s–1 (at the redshift of the clusters) and within an area of 1 h –1 72 Mpc radius (excluding the core). To investigate the presence or absence of a true enhancement of luminous X-ray AGNs in the cluster area, we also derived the corresponding optical galaxy overdensities, using a suitable range of r-band magnitudes. We always find the latter to be significantly higher (and only in two cases roughly equal) with respect to the corresponding X-ray overdensities. Over the whole cluster sample, the mean X-ray point-source overdensity is a factor of ~4 less than that corresponding to bright optical galaxies, a difference which is significant at a >0.995 level, as indicated by an appropriate student’s t-test. We conclude that the triggering of luminous X-ray AGNs in rich clusters is strongly suppressed. Furthermore, searching for optical Sloan Digital Sky Survey counterparts of all the X-ray sources, associated with our clusters, we found that about half appear to be background QSOs, while others are background and foreground AGNs or stars. The true overdensity of X-ray point sources, associated with the clusters, is therefore even smaller than what our statistical approach revealed.