  The MUonE experiment aims at a precision measurement of the hadronic vacuum
polarization contribution to the muon $g-2$, via elastic muon-electron
scattering. Since the current muon $g-2$ anomaly hints at the potential
existence of new physics (NP) related to the muon, the question then arises as
to whether the measurement of hadronic vacuum polarization in MUonE could be
affected by the same NP as well. In this work, we address this question by
investigating a variety of NP explanations of the muon $g-2$ anomaly via either
vector or scalar mediators with either flavor-universal, non-universal or even
flavor-violating couplings to electrons and muons. We derive the corresponding
MUonE sensitivity in each case and find that the measurement of hadronic vacuum
polarization at the MUonE is not vulnerable to any of these NP scenarios.
