  In this study, we calculate the sensitivity reach on the vector leptoquark
(LQ) $U_1$ at the experiments proposed in Forward Physics Facility (FPF),
including FASER$\nu$, FASER$\nu2$, FLArE (10 tons), and FLArE (100 tons) using
the neutrino-nucleon scattering ($\nu N \rightarrow \nu N'$ and $\nu N
\rightarrow l N'$). We cover a wide mass range of $10^{-3}$ GeV $\leq
M_{LQ}\leq 10^4$ GeV. The new result shows that the FLArE (100 tons) offers the
best sensitivity to the LQ model. The sensitivity curves for all the
experiments follow a similar pattern with weakened sensitivities with the
increment of the LQ mass. We combine the sensitivities obtained from the
neutral- and charged-current interactions of the neutrinos.
