  New physics contributions to the $Z$ penguin are revisited in the light of
the recently-reported discrepancy of the direct CP violation in $K\to\pi\pi$.
Interference effects between the standard model and new physics contributions
to $\Delta S = 2$ observables are taken into account. Although the effects are
overlooked in the literature, they make experimental bounds significantly
severer. It is shown that the new physics contributions must be tuned to
enhance $\mathcal{B}(K_L \to \pi^{0} \nu \bar{\nu})$, if the discrepancy of the
direct CP violation is explained with satisfying the experimental constraints.
The branching ratio can be as large as $6 \times 10^{-10}$ when the
contributions are tuned at the 10 % level.
