  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.
