Abstract By combining the REFLEX and NORAS cluster data sets with the NVSS radio catalogue, we obtain a sample of 145, z < 0.3, X-ray-selected clusters brighter than 3 × 10−12 erg s−1 cm−2 that show a central radio emission above 3 mJy. For virial masses Mvir≲ 1014.5 M⊙, 11 clusters out of 12 (corresponding to 92 per cent of the systems) are inhabited by a central radio source. This fraction decreases with higher masses as ∝M−0.4vir. If this decrease is a selection effect, it suggests that the majority of X-ray-selected clusters host in their centre a radio source brighter than ∼1020 W Hz−1 sr−1. A division of the sample into clusters harbouring either point-like or an extended radio-loud active galactic nucleus (AGN) reveals that the steepening of the LX–T relation for low-temperature clusters is strongly associated with the presence of central radio objects with extended jets and/or lobe structures. In the latter case, LX∝T4 while for point-like sources one recovers an approximately self-similar relation LX∝T2.3. Monte Carlo simulations show that the steepening of the LX–T relation is not caused by clusters being underluminous in the X-ray band, but rather by overheating, most likely caused by the interplay between the extended radio structures and the intracluster medium. In the case of low-mass systems, we also find a tight correlation between radio luminosity and cluster temperature. The effects of the central radio source on the thermal state of a cluster become less important with increasing cluster mass. The presence of radio sources with extended structures (61, corresponding to ∼42 per cent of the sample) is enhanced in X-ray luminous clusters with respect to ‘field’ radio-loud AGN. Furthermore, we find that the luminosity distribution of the cluster radio population differs from that of all radio sources, as there is a deficit of low-luminosity (LR≲ 1022 W Hz−1 sr−1) objects, while the number of high-luminosity ones is boosted. The net effect on the radio luminosity function of radio galaxies associated with cluster centres is of a flattening at all luminosities LR≲ 1024 W Hz−1 sr−1.