  Recent calculations have pointed to a 2.8 $\sigma$ tension between data on
$\epsilon^{\prime}_K / \epsilon_K$ and the standard-model (SM) prediction.
Several new physics (NP) models can explain this discrepancy, and such NP
models are likely to predict deviations of $\mathcal{B}(K\to \pi \nu
\overline{\nu})$ from the SM predictions, which can be probed precisely in the
near future by NA62 and KOTO experiments. We present correlations between
$\epsilon^{\prime}_K / \epsilon_K$ and $\mathcal{B}(K\to \pi \nu
\overline{\nu})$ in two types of NP scenarios: a box dominated scenario and a
$Z$-penguin dominated one. It is shown that different correlations are
predicted and the future precision measurements of $K \to \pi \nu
\overline{\nu}$ can distinguish both scenarios.
