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 $Λ^{k}$ and $Λ^{\leq k}$ 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 $T(Λ^{\leq k}) = Λ^{\leq k}$ or bijective linear maps satisfying $T(Λ^{\leq k}) \subseteq Λ^{\leq k}$. Moreover, we show that if $\mathbb{F}$ is an infinite field, then $Λ^{k}$ is Zariski dense in $Λ^{\leq k}$ and apply this to describe such bijective linear maps satisfying $T(Λ^{k}) \subseteq Λ^{k}$.