Let $R$ be a commutative ring with identity. If $M$ is a finitely generated $R$-module, then $\textrm{Ass}(\textrm{Hom}_{R}(M,N))\subseteq\textrm{Supp}(M)\cap\textrm{Ass}(N)$ for every $R$-module $N$ and the converse inclusion holds if $R$ is Noetherian. In this case we conclude that $\textrm{Supp}(\textrm{Hom}_{R}(M,N))\subseteq\textrm{Supp}(M)$.
My question is if there exists any relation between $\textrm{Supp}(\textrm{Hom}_{R}(M,N))$ and $\textrm{Supp}(M)$ dropping the condition that $M$ is finitely generated. It can also be assumed that $N$ is the minimal injective cogenerator of $R$. Thank you.