  In this paper, we study operator-theoretic properties (boundedness and
compactness) of Hankel operators on the Fock-sobolev spaces $ \mathscr{F}^{p,m}
$ in terms of $ \mathcal{BMO}_r^p $ and $ \mathcal{VMO}_r^p $ spaces,
respectively, for a non-negative integers $ m $, $ 1 \leq p < \infty $ and $ r
> 0 $. Along the way, we also study Berezin transform of Hankel operators on $
\mathscr{F}^{p,m} $. The results in this article are analogous to Zhu's
characterization and Per\"al\"a's characterization of bounded and compact
Hankel operators on the Bergman spaces of unit disc and the weighted Fock
spaces, respectively.
