The dying radio sources represent a very interesting and largely unexplored stage of the active galactic nucleus (AGN) evolution. They are considered to be very rare, and almost all of the few known ones were found in galaxy clusters. However, considering the small number detected so far, it has not been possible to draw any firm conclusions about their X-ray environment. We present X-ray observations performed with the Chandra satellite of the three galaxy clusters Abell 2276, ZwCl 1829.3+6912, and RX J1852.1+5711, which harbor at their center a dying radio source with an ultra-steep spectrum that we recently discovered. We analyzed the physical properties of the X-ray emitting gas surrounding these elusive radio sources. We determined the global X-ray properties of the clusters, derived the azimuthally averaged profiles of metal abundance, gas temperature, density, and pressure. Furthermore, we estimated the total mass profiles. The large-scale X-ray emission is regular and spherical, suggesting a relaxed state for these systems. Indeed, we found that the three clusters are also characterized by significant enhancements in the metal abundance and declining temperature profiles toward the central region. For all these reasons, we classified RX J1852.1+5711, Abell 2276, and ZwCl 1829.3+6912 as cool-core galaxy clusters. ; Comment: Accepted for publication in Astronomy and Astrophysics, 15 pages, 8 figures