  We study the impact of non-standard neutrino interactions in the context of a
new gauge boson $Z'$ in neutral-current deep-inelastic scattering performed in
ForwArd Search ExpeRiment-$\nu$ (FASER$\nu$) and in monojet production at the
Large Hadron Collider (LHC). We simulate the neutral-current deep-inelastic
neutrino-nucleon scattering $\nu N \rightarrow \nu N$ at FASER$\nu$ in the
presence of an additional $Z'$ boson, and estimate the anticipated
sensitivities to the gauge coupling in a wide range of $Z'$ mass. At the LHC,
we study the effect of $Z'$ on monojet production, which can be enhanced in
regions with large missing transverse momenta. We then use the recent results
from ATLAS with an integrated luminosity of 139 fb$^{-1}$ to improve the limits
on the gauge coupling of $Z'$. We interpret such limits on $Z'$ gauge couplings
as bounds on effective non-standard neutrino interactions. We show that the
FASER$\rm \nu$ and the LHC results cover the medium and high energy scales,
respectively, and complement one another.
