  We study the prospects of using transitions from the ytterbium ground state
to metastable states $^3{\rm P}^{\rm o}_2$ at $E=19\,710.388~$cm$^{-1}$ and
$4f^{13}5d6s^2\,(J=2)$ at $E=23\,188.518~$cm$^{-1}$ as clock transitions in an
optical lattice clock. Having more than one clock transition in Yb could
benefit the search for new physics beyond the Standard Model via studying the
non-linearity of King's plot or the time-variation of the ratio of the
frequencies of two clock transitions. We calculate the lifetime of the states,
relevant transition amplitudes, systematic shifts, and the sensitivities of the
clock transitions to a variation of the fine structure constant and to the
gravitational potential. We find that both transitions can probably support
ultra-high accuracy, similar to what is already achieved for the $^1$S$_0$ -
$^3$P$^{\rm o}_0$ clock transition.
