  In this article, we first prove the existence of an invariant subspace for a
norm-attaining $\ast$-paranormal operator. Then give a representation for
$\ast$-paranormal operators in the closure of absolutely norm-attaining
operators and further study a few sufficient conditions for the normality of
such operators. Finally, we discuss Toeplitz and Hankel $\ast$-paranormal
operators in the closure of absolutely norm-attaining operators on the Hardy
space.
