Abstract: The present and upcoming neutrino experiments have ushered in neutrino precision era which makes it imperative to study the possibility of subdominant Non-Standard neutrino Interaction (NSI) with matter fields. The NSI between neutrinos and matter field involving a heavy mediator can be described by a four-Fermi effective potential. While the vector part of such interactions has received tremendous attention in the literature both from theoretical and experimental side, the axial coupling needs more study.  In this talk, I will review the existing bounds and then discuss how the future  state-of-the-art long baseline neutrino experiment, DUNE can probe axial NSI. I will then present a model that leads to axial NSI with couplings that can be probed by DUNE. As a bonus, the model has a dark matter candidate which can be discovered by spin dependent direct  dark matter detection experiments.