  Recently the neutrino experiments ANTARES and IceCube have released new
constraints to the non-standard neutrino interaction (NSI) parameter
$\epsilon^d_{\mu\tau}$ (flavor off-diagonal). These new constraints are
stronger than those obtained from a combination of COHERENT and neutrino
oscillation data. In the light of the recent constraints from ANTARES and
IceCube data on the NSI parameter $\epsilon^d_{\mu\tau}$, in this work, we
study the new physics implications on the parameter space of a simplified
$Z^\prime$ model with lepton flavor violating ($\mu\tau$) couplings. For a
$Z^\prime$ boson with a mass heavier than the $\tau$ lepton, our results show
that ANTARES and IceCube can provide additional constraints to such a new
physics scenario with $\mu\tau$ couplings, when compared to bounds from
low-energy flavor physics. Moreover, these neutrino experiments can exclude a
similar region than ATLAS experiment, showing the potential to provide
complementary information to the one obtained from direct searches at the Large
Hadron Collider. The impact of the expected sensitivity at DUNE and Belle II
experiments is also studied.
