Towards experimental confirmations of the type-I seesaw mechanism, we explore a prospect of discovering the heavy Majorana right-handed neutrinos (RHNs) from a resonant production of a new massive gauge boson $(Z^\prime)$ and its subsequent decay into a pair of RHNs $(Z^\prime \to NN)$ at the High Luminosity Large Hadron Collider (HL-LHC). Recent simulation studies have shown that the discovery of the RHNs through this process is promising in the future. However, the current LHC data very severely constrains the production cross section of the $Z^\prime$ boson into a dilepton final states, $(pp \to Z^\prime \to \ell^+ \ell^-,~\ell=e/\mu)$. Extrapolating the current bound to the future, we find that a significant enhancement of the branching ratio BR$(Z^\prime \to NN)$ over BR$(Z^\prime \to \ell^+ \ell^-)$ is necessary for the future discovery of RHNs. As a well-motivated simple extension of the standard model (SM) to incorporate the $Z^\prime$ boson and the type-I seesaw mechanism, we consider the minimal $U(1)_X$ model. We point out that this model can yield a significant enhancement up to BR$(Z^\prime \to NN)$/BR$(Z^\prime \to \ell^+ \ell^-) \simeq 5$ (per generation).With such an enhancement and a realistic model-parameter choice to reproduce the neutrino oscillation data, we conclude that the possibility of discovering RHNs with, for example, a $3000$~fb$^{-1}$ luminosity implies that the $Z^\prime$ boson will be discovered with a luminosity of $853$~fb$^{-1}$ ($626$~fb$^{-1}$) for the normal (inverted) hierarchy of the light neutrino mass pattern.