We present X-ray observations of the nuclear region of 25 Fanaroff-Riley I radio galaxies from the 3CRR and B2 catalogs, using data from the Chandra and XMM archives. We find the presence of a X-ray Central Compact Core (CCCX) in 13/25 sources, in 3/25 sources the detection of a CCCX is uncertain, while in the remaining 9/25 sources no CCCX is found. All the sources are embedded in a diffuse soft X-ray component, generally on kpc-scales, which is in agreement with the halo of the host galaxy and/or with the intracluster medium. The X-ray spectra of the cores are described by a power law with photon indices Gamma=1.1 - 2.6. In 8 sources excess absorption over the Galactic value is detected, with rest-frame column densities N_H^z ~ 10^20 - 10^21 cm^-2; thus, we confirm the previous claim based on optical data that most FRI radio galaxies lack a standard optically-thick torus. We find significant correlations between the X-ray core luminosity and the radio and optical luminosities, suggesting that at least a fraction of the X-ray emission originates in a jet; however, the origin of the X-rays remains ambiguous. If the X-ray emission is entirely attributed to an isotropic, accretion-related component, we find very small Eddington ratios, L_bol/L_Edd ~ 10^-3 - 10^-8, and we calculate the radiative efficiency to be eta ~ 10^-2 - 10^-6, based on the Bondi accretion rates from the spatial analysis. This suggests that radiatively inefficient accretion flows are present in the cores of low-power radio galaxies.
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