The phonon magnetochiral effect consists of a nonreciprocity in the velocity or attenuation of acoustic waves when they propagate parallel and antiparallel to an external magnetic field. The first experimental observation of this effect in the bulk has been reported recently in a chiral magnet and ascribed to the hybridization between acoustic phonons and chiral magnons. Here, we predict a potentially measurable phonon magnetochiral effect of electronic origin in chiral Weyl semimetals. Caused by the Berry curvature and the orbital magnetic moment, this effect is enhanced for longitudinal phonons by the chiral anomaly.