  Using the effective Lagrangian formalism, the anomalous magnetic and electric
dipole moments of the tau-lepton in the $\mu^{+}\mu^{-} \rightarrow
\mu^{+}\gamma^{*} \gamma^{*} \mu^{-}\rightarrow \mu^{+}\tau \bar{\tau}\mu^{-}$
process at the future muon colliders at the $\sqrt{s}=1.5, 3$ and $6$ TeV are
investigated. In addition, the bounds at the $95\%$ confidence level on the
dipole moments of the tau-lepton using different integrated luminosities are
estimated. It is shown that the $\mu^{+}\mu^{-} \rightarrow \mu^{+}\gamma^{*}
\gamma^{*} \mu^{-}\rightarrow \mu^{+}\tau \bar{\tau}\mu^{-}$ process leads to a
remarkable improvement in the existing experimental bounds on the anomalous
magnetic and electric dipole moments of the tau-lepton.
