  Sterile neutrinos with mass in the eV-scale and large mixings of order
$\theta_0\simeq 0.1$ could explain some anomalies found in short-baseline
neutrino oscillation data. Here, we revisit a neutrino portal scenario in which
eV-scale sterile neutrinos have self-interactions via a new gauge vector boson
$\phi$. Their production in the early Universe via mixing with active neutrinos
can be suppressed by the induced effective potential in the sterile sector. We
study how different cosmological observations can constrain this model, in
terms of the mass of the new gauge boson, $M_\phi$, and its coupling to sterile
neutrinos, $g_s$. Then, we explore how to probe part of the allowed parameter
space of this particular model with future observations of the diffuse
supernova neutrino background by the Hyper-Kamiokande and DUNE detectors. For
$M_\phi \sim 5-10$~keV and $g_s \sim 10^{-4}-10^{-2}$, as allowed by
cosmological constraints, we find that interactions of diffuse supernova
neutrinos with relic sterile neutrinos on their way to the Earth would result
in significant dips in the neutrino spectrum which would produce unique
features in the event spectra observed in these detectors.
