Lepton flavor violation (LFV) is known to be a suitable avenue to look for physics beyond the Standard Model. With experiments like LHCb and Belle II running in the flavor sector, we expect to get a lot of data that will help search for the hint of new physics. There exist few anomalies in the quark sector, although there is no clear evidence of new physics. On the other hand, the observation of neutrino oscillation has opened a new window to particle physics, indicating physics beyond the Standard Model. In our work, we study the charged LFV $\mu$ decays such as $\mu\rightarrow e\gamma$, $\mu \rightarrow eee$, and $(\mu - e)_{\text{Ti}}$ with a vector leptoquark ($U_3$) by considering the constraints from non-standard neutrino interaction (NSI) sector parameter $\epsilon_{e\mu}$. We consider that these NSIs are attributed to the presence of leptoquarks to account for the difference in the experimental observations of $\delta_{CP}$ measurement by NOvA and T2K. We obtained the branching ratios with uncertainties for three decay modes: $(\mu \rightarrow e \gamma) \leq 10^{-18}$, $(\mu \rightarrow eee) \leq 10^{-21}$ and $(\mu \rightarrow e)_{\text{Ti}} \leq 10^{-19}$.
Our results show an improvement in the current limits, which can be explored in future experiments.