  Let ${\rm Mat}_n(\mathbb{F})$ denote the set of square $n\times n$ matrices
over a field $\mathbb{F}$ of characteristic different from two. The permanental
rank ${\rm prk}\,(A)$ of a matrix $A \in{\rm Mat}_{n}(\mathbb{F})$ is the size
of the maximal square submatrix in $A$ with nonzero permanent. By $\Lambda^{k}$
and $\Lambda^{\leq k}$ we denote the subsets of matrices $A \in {\rm
Mat}_{n}(\mathbb{F})$ with ${\rm prk}\,(A) = k$ and ${\rm prk}\,(A) \leq k$,
respectively. In this paper for each $1 \leq k \leq n-1$ we obtain a complete
characterization of linear maps $T: {\rm Mat}_{n}(\mathbb{F}) \to {\rm
Mat}_{n}(\mathbb{F})$ satisfying $T(\Lambda^{\leq k}) = \Lambda^{\leq k}$ or
bijective linear maps satisfying $T(\Lambda^{\leq k}) \subseteq \Lambda^{\leq
k}$. Moreover, we show that if $\mathbb{F}$ is an infinite field, then
$\Lambda^{k}$ is Zariski dense in $\Lambda^{\leq k}$ and apply this to describe
such bijective linear maps satisfying $T(\Lambda^{k}) \subseteq \Lambda^{k}$.
