Let $S\subset\mathbb Z_n$ be a finite cyclic group of order $n\geq1$. Assume that $0\not\in S$ and $-S=\{-s:s\in S\}=S$. The circulant graph $G=\operatorname{Cir}(n,S)$ is the undirected graph having the vertex set $V(G)=\mathbb Z_n$ and edge set $E(G)=\{ab:a,b\in\mathbb Z_n, \ a-b\in S\}$. Let $\mathcal D$ be a set of positive, proper divisors of the integer $n$. We characterize certain strongly regular integral circulant graphs with energy $2n(1-1/d)$ for a fixed $d\in\mathcal D$, $d>1$.