  We study compactness of product of Toeplitz operators with symbols continuous
on the closure of the polydisc in terms of behavior of the symbols on the
boundary. For certain classes of symbols $f$ and $g$, we show that $T_fT_g$ is
compact if and only if $fg$ vanishes on the boundary. We provide examples to
show that for more general symbols, the vanishing of $fg$ on the whole polydisc
might not imply the compactness of $T_fT_g$. On the other hand, the reverse
direction is closely related to the zero product problem for Toeplitz operators
on the unit disc, which is still open.
