  High lithium-7 ($\mathrm{^7Li}$) abundances in giants are indicative of
non-standard physical processes affecting the star. Mechanisms that could
produce this signature include contamination from an external source, such as
planets, or internal production and subsequent mixing to the stellar surface.
However, distinguishing between different families of solutions has proven
challenging, and there is no current consensus model that explains all the
data. The lithium-6 ($\mathrm{^6Li}$) abundance may be a potentially important
discriminant, as the relative $\mathrm{^6Li}$ and $\mathrm{^7Li}$ abundances
are expected to be different if the enrichment were to come from internal
production or from engulfment. In this work, we model the $\mathrm{^6Li}$ and
$\mathrm{^7Li}$ abundances of different giants after the engulfment of a
substellar mass companion. Given that $\mathrm{^6Li}$ is more strongly affected
by Galactic chemical evolution than $\mathrm{^7Li}$, $\mathrm{^6Li}$ is not a
good discriminant at low metallicities, where it is expected to be low in both
star and planet. For modeled metallicities ([Fe/H]$>-0.5$), we use a "best
case" initial $\mathrm{^6Li/^7Li}$ ratio equal to the solar value.
$\mathrm{^6Li}$ increases significantly after the engulfment of a companion.
However, at metallicities close to solar and higher, the $\mathrm{^6Li}$ signal
does not last long in the stellar surface. As such, detection of surface
$\mathrm{^6Li}$ in metal-rich red giants would most likely indicate the action
of a mechanism for $\mathrm{^6Li}$-enrichment other than planet engulfment. At
the same time, $\mathrm{^6Li}$ should not be used to reject the hypothesis of
engulfment in a $\mathrm{^7Li}$-enriched giant or to support a particular
$\mathrm{^7Li}$-enhancement mechanism.
