The lepton flavor-violating ${Z}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\tau}\ensuremath{\mu}$ decay is studied in the context of several extended models that predict the existence of a new gauge boson named ${Z}^{\ensuremath{'}}$. A calculation of the strength of the lepton flavor-violating ${Z}^{\ensuremath{'}}\ensuremath{\mu}\ensuremath{\tau}$ coupling is presented by using the most general renormalizable Lagrangian that includes lepton flavor violation. We used the experimental value of the muon magnetic dipole moment to bound this coupling, from which the $\mathrm{Re}({\ensuremath{\Omega}}_{L\ensuremath{\mu}\ensuremath{\tau}}{\ensuremath{\Omega}}_{R\ensuremath{\mu}\ensuremath{\tau}}^{*})$ parameter is constrained, and it was found that $\mathrm{Re}({\ensuremath{\Omega}}_{L\ensuremath{\mu}\ensuremath{\tau}}{\ensuremath{\Omega}}_{R\ensuremath{\mu}\ensuremath{\tau}}^{*})\ensuremath{\sim}{10}^{\ensuremath{-}2}$ for a ${Z}^{\ensuremath{'}}$ boson mass of 2 TeV. We also employed the experimental restrictions over the $\ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\mu}\ensuremath{\gamma}$ and $\ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\mu}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ processes in the context of several models that predict the existence of the ${Z}^{\ensuremath{'}}$ gauge boson to bound the mentioned coupling. The most restrictive bounds come from the calculation of the three-body decay. For this case, it was found that the most restrictive result is provided by a vectorlike coupling, denoted as $|{\ensuremath{\Omega}}_{\ensuremath{\mu}\ensuremath{\tau}}{|}^{2}$, for the ${Z}_{\ensuremath{\chi}}$ case, finding around ${10}^{\ensuremath{-}2}$ for a ${Z}^{\ensuremath{'}}$ boson mass of 2 TeV. We used this information to estimate the branching ratio for the ${Z}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\tau}\ensuremath{\mu}$ decay. According to the analyzed models the least optimistic result is provided by the sequential $Z$ model, which is of the order of ${10}^{\ensuremath{-}2}$ for a ${Z}^{\ensuremath{'}}$ boson mass around 2 TeV.