  The muonic vacuum polarization contribution to the $g$-factor of the electron
bound in a nuclear potential is investigated theoretically. The electric as
well as the magnetic loop contributions are evaluated. We found these muonic
effects to be observable in planned trapped-ion experiments with light and
medium-heavy highly charged ions. The enhancement due to the strong Coulomb
field boosts these contributions much above the corresponding terms in the
free-electron $g$-factor. Due to their magnitude, muonic vacuum polarization
terms are also significant in planned determinations of the fine-structure
constant from the bound-electron $g$-factor.
