  The WL-rank of a digraph $\Gamma$ is defined to be the rank of the coherent
configuration of $\Gamma$. The WL-dimension of $\Gamma$ is defined to be the
smallest positive integer $m$ for which $\Gamma$ is identified by the
$m$-dimensional Weisfeiler-Leman algorithm. We classify the Deza circulant
graphs of WL-rank $4$. In additional, it is proved that each of these graphs
has WL-dimension at most $3$. Finally, we establish that some families of Deza
circulant graphs have WL-rank $5$ or $6$ and WL-dimension at most $3$.
