  Let $G$ be a locally compact group, $A_p (G)$ be the Herz algebra of $G$
associated with $1 <p< \infty$. We show that $A_p (G)$ is Arens regular if and
only if $G$ is a discrete group and for each countable subgroup $H$ of $G$,
$A_p (H)$ is Arens regular. In the case $G$ is a countable discrete group we
investigate the relations between Arens regularity of $A_p (G)$ and the
iterated limit condition. We consider the problem of Arens regularity of $l^1
(G)$ as a subspace of $A_p (G)$. A few related results when the unit ball of
$(l^1 (G),.,A_p(G))$ is bounded under $\|.\|_1$-norm are also determined.
