Using new, homogeneous, long-slit spectroscopy in the wavelength range from ~0.35 to $\sim 1 \, \mu$m, we study radial gradients of optical and near-infrared (NIR) initial mass function (IMF)-sensitive features along the major axis of the bulge of M31, out to a galactocentric distance of ~200 arcsec (~800 pc). Based on state-of-the-art stellar population synthesis models with varying Na abundance ratio, we fit a number of spectral indices, from different chemical species (including TiO's, Ca, and Na indices), to constrain the low-mass (≲0.5 M⊙) end slope (i.e. the fraction of low-mass stars) of the stellar IMF, as a function of galactocentric distance. Outside a radial distance of ~10 arcsec, we infer an IMF similar to a Milky Way-like distribution, while at small galactocentric distances, an IMF radial gradient is detected, with a mildly bottom-heavy IMF in the few inner arcsec. We are able to fit Na features (both NaD and $\rm Na\,{\small I}8190$), without requiring extremely high Na abundance ratios. $\rm [Na/Fe]$ is ~0.4 dex for most of the bulge, rising up to ~0.6 dex in the innermost radial bins. Our results imply an overall, luminosity-weighted, IMF and mass-to-light ratio for the M31 bulge, consistent with those for a Milky Way-like distribution, in contrast to results obtained, in general, for most massive early-type galaxies.