  Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module.
The aim of this paper is to extend the notion of quasi $J$-ideals of
commutative rings to quasi $J$-submodules of modules. We call a proper
submodule $N$ of $M$ a quasi $J$-submodule if whenever $r\in R$ and $m\in M$
such that $rm\in N$ and $r\notin(J(R)M:M)$, then $m\in M$-$rad(N)$. We present
various properties and characterizations of this concept (especially in
finitely generated faithful multiplication modules). Furthermore, we provide
new classes of modules generalizing presimplifiable modules and justify their
relation with (quasi) $J$-submodules. Finally, for a submodule $N$ of $M$ and
an ideal $I$ of $R$, we characterize the quasi $J$-ideals of the idealization
ring $R(+)M$.
