We analyze the signal-to-noise ratio for a relic background of scalar gravitational radiation composed of massive, nonrelativistic particles, interacting with the monopole mode of two resonant spherical detectors. We find that the possible signal is enhanced with respect to the differential mode of the interferometric detectors. This enhancement is due to (a) the absence of the signal suppression, for nonrelativistic scalars, with respect to a background of massless particles, and (b) for flat enough spectra, a growth of the signal with the observation time faster than for a massless stochastic background.