Narrow-band Hα+[NII] imaging and long-slit spectroscopy at intermediate resolution from [NeV] lambda3426 to [SIII] lambda9532 of the LINER galaxies NGC 4579 and 6500 are analysed. Both galaxies show circumnuclear emission, extending 2kpc from the nucleus of NGC 4579 in a loop structure and 1 kpc from the nucleus of NGC 6500 in a bright knot to the north-east, and as diffuse emission to the east. The nuclei and the circumnuclear regions in both galaxies show strong low-ionization lines. The nuclear line ratios [NII] lambda6584/Hα, [OI] lambda6300/Hα and [OII] lambda3727/[OIII] lambda5007 are higher than 0.6, 0.3 and 1, respectively, fitting the definition criteria of a LINER. The circumnuclear gas is kinematically perturbed, showing radial motions; in particular, in NGC 6500 it is compatible with an outflow wind which interacts with the higher density gas in the plane of the galaxy. Comparison of the observed line ratios, particularly [SIII]/[SII] and [OII] lambda7325/Hα, with those of photoionization and shock heating models, indicates that the main excitation mechanism in the extended emission and in the nuclei is due to shocks.