  Neutral hydrogen (HI) intensity mapping traces the large-scale distribution
of matter in the Universe and therefore should correlate with the gamma-ray
emission originated from particle dark matter annihilation or from active
galactic nuclei and star-forming galaxies, since the related processes occur in
the same cosmic structures hosting HI. In this paper, we derive the
cross-correlation signal between the brightness temperature of the 21-cm line
emission of the HI spin-flip transition in the Universe and the unresolved
gamma-ray background. Specifically, we derive forecasts for the
cross-correlation signal by focussing on the opportunities offered by the
combination of the Fermi-Large Area Telescope (LAT) gamma-ray sensitivity with
the expectations of the HI intensity mapping measurements from future radio
telescopes, for which we concentrate on the Square Kilometre Array (SKA) and
MeerKAT, one of its precursors. We find that the combination of MeerKAT with
the current Fermi-LAT statistics has the potential to provide a first hint of
the cross-correlation signal originated by astrophysical sources, with a
signal-to-noise ratio (SNR) of 3.7. With SKA Phase 1 and SKA Phase 2, the SNR
is predicted to increase up to 5.7 and 8.2, respectively. The bounds on dark
matter properties attainable with SKA combined with the current statistics of
Fermi-LAT are predicted to be comparable to those obtained from other
techniques able to explore the unresolved components of the gamma-ray
background. The enhanced capabilities of SKA Phase 2, combined with a future
generation gamma-ray telescope with improved specifications, can allow us to
investigate the whole mass window for weakly interacting massive particles up
to the TeV scale.
