Let $R$ be a ring. For two fixed positive integers $m$ and $n$, an $R$-module $M$ is called {\it\bfseries $(m,n)$-quasi-injective} if each $R$-homomorphism from an $n$-generated submodule of $M^{m}$ to $M$ extends to one from $M^{m}$ to $M$. It is showed that $M_R$ is $(m,n)$-quasi-injective if and only if the right $R^{n\times n}$-module $M^{m\times n}$ is principally quasi-injective. Many properties of $(m,n)$-injective rings and principally quasi-injective modules are extended to these modules. Moreover, some properties of $(m,n)$-quasi-injective Kasch modules are investigated. In particular, some other well-known results are also obtained.