  We explore the possibility to explain a bosonic dark matter candidate with a
gauge singlet inside the loop to generate the neutrino mass matrix at two-loop
level. The mass matrix is suppressed by a small mixing that comes from the
bound on {direct detection experiments of the dark matter, and equivalent of
the three-loop neutrino model due to the small mixing between neutral inert
bosons. Here, our setup is} the Zee-Babu type scenario with $Z_3$ discrete
symmetry, in which we consider the neutrino oscillation data, lepton flavor
violations, muon $g-2$, $\mu-e$ conversion rate, lepton flavor-changing and
conserving $Z$ boson decay and bosonic dark matter candidate.
