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 ~\cite{nova,t2k}. In our study, we obtained the branching ratios with uncertainties for three decay modes:
\begin{eqnarray*}
\mathcal{B}(\mu \rightarrow e \gamma) \leq 4.6^{+3.8}_{-2.6} \times 10^{-18},\\ 
\mathcal{B}(\mu \rightarrow eee) \leq 9.57^{+7.43}_{-5.37} \times 10^{-21},\\
\mathcal{B}(\mu \rightarrow e)_{\text{Ti}} \leq 6.89^{+5.61}_{-3.89} \times 10^{-19}.
\end{eqnarray*}
Our results show an improvement in the current limits, which can be explored in future experiments.