Let $D$ be an integral domain and $\overline{D}$ be the integral closure of $D$. Is there any condition equivalent to the property that there exists $0\neq x\in D$ such that $x\overline{D}\subseteq D$?
Note that if $D=\overline{D}$, then the above property always holds.
If $D$ is noetherian, this is equivalent to the statement that $D \subset \overline D$ is a finite extension.
Clearly, if that extension is finite, you just take $x$ to be the product of all denominators among generators of $\overline D$. No noetherian assumption needed.
On the other hand, if $x\overline D \subset D$ holds, you have $\overline D \subset \frac{1}{x}D$ and the latter is a noetherian $D$-module if $D$ is noetherian.