We present new K-band spectroscopy for the giant elliptical galaxy M87 in the Virgo cluster, taken with the Large Binocular Telescope Utility Camera in the Infrared (LUCI) spectrograph at the Large Binocular Telescope (LBT). The new data are used to study line strengths of K-band absorption features from different chemical species, namely Fe, Mg, Ca, Na, and CO, as a function of galactocentric distance, out to ∼40″ from the center (about half of the galaxy effective radius). The radial trends of spectral indices are compared to those for the bulge of M31, observed with the same instrument. For M87, most K-band indices exhibit flat radial profiles, with the exception of NaI2.21, which decreases outward, with a negative radial gradient. Significant offsets are found between indices for M87 and those for the bulge of M31, the latter having weaker line strengths for almost all features, but Fe and Ca, for which we find similar trends in both systems. We find that the behavior of CO features – most prominent in giant stars – is difficult to explain, consistent with previous results for the central regions of massive galaxies. In particular, the CO indices are stronger in M87 than M31, and do not exhibit significant radial gradients in M87, despite its IMF being bottom heavier than M31 especially in its central region. Predictions of state-of-the-art stellar population models, based on results from the optical spectral range, are able to match only the Na and Ca indices of M87, while a significant mismatch is found for all other indices. This shows that state-of-the-art stellar population models should be improved significantly in order to provide reliable constraints on the stellar population content of galaxies in the near-infrared spectral range.