  The increasingly precise neutrino experiments raise the hope for searching
for new physics through studying the impact of Neutral Current (NC)
Non-Standard Interactions (NSI) of neutrinos with matter fields. Neutrino
oscillation experiments along with the Elastic Coherent $\nu$ Nucleus
Scattering (CE$\nu$NS) experiments already set strong bounds on all the flavor
elements of the "vector" NC NSI. However, "axial" NC NSI can hide from these
experiments. We show how a DUNE-like experiment can probe these couplings by
studying NC Deep Inelastic Scattering (DIS) events. We find that strong bounds
can be set on the axial NC NSI of neutrinos with the $u$, $d$, and $s$ quarks.
We show that using both the near and far detectors, a DUNE-like experiment can
significantly improve the present bounds on all the flavor elements.
