Near-infrared spectroscopic data for the five Seyfert galaxies with jet-gas interaction Mrk 348, Mrk 573, Mrk 1066, NGC 7212, and NGC 7465, taken with the LIRIS near-infrared camera/spectrometer at the William Herschel Telescope (WHT) are reported. The long-slit spectra reveal the characteristic strong emission lines of this type of objects. Many forbidden transitions and hydrogen recombination lines are employed here to study the excitation and ionization mechanisms that are dominating the narrow-line region emission of these objects, that is affected by the radio-jet interaction. Several absorption features are also detected in the H and K bands of these galaxies, allowing us to identify the spectral types that are producing them. We find that the continuum can be reproduced by a combination of late-type stellar templates plus a Blackbody component associated to host dust, mainly contributing to the K band emission. The detection of the permitted O I and Fe II lines and broad components of the hydrogen recombination lines in the spectra of Mrk 573 and NGC 7465 allows the reclassification of these two galaxies that are not canonical Type-2 Seyferts: Mrk 573 is confirmed to be an obscured Narrow-line Seyfert 1 and NGC 7465 is revealed for the first time as a Type-1 LINER through its near-infrared spectrum. ; Comment: 44 pages, 16 figures. Accepted for publication in ApJ