Abstract. We present the first near-infrared (K -band) homogeneous observations of a complete sub-sample of the 3CR radio catalogue comprising all High Excitation Galaxies (HEGs) at z < 0.3. After showing that the surface brightness decomposition technique to measure central point-like sources is affected by significant uncertainties for the objects in the studied sample, we present a new, more accurate method based on the R − K color profile. Via this method we find a substantial nuclear K -band excess in all but two HEGs -most likely directly associated with their nuclear emission -and we measure the corresponding 2.12 µm nuclear luminosities. Within the frame-work of the unification scheme for radio-loud active galactic nuclei, it appears that obscuration alone is not able to account for the different nuclear properties of the majority of the HEGs and Broad Line Radio Galaxies (BLRGs), and also scattering of the (optically) hidden nuclear light from a compact region must be invoked. More precisely, for ∼70% of the HEGs the observed point-like optical emission is dominated by the scattered component, while in the K -band both scattered and direct light passing through the torus contribute to the observed nuclear luminosity. The estimated fraction of scattered light ranges from a few tenths to a few percent, while the torus extinction is between 15 < A V, torus < 50 mag with only a few exceptions with lower obscuration.