It seems to me the references in this Mathematics - Stack Exchange answer contain the requested information.
EDIT. Here is an excerpt from Hyman Bass's book Algebraic K-Theory, W. A. Benjamin (1968), p. 54:
Exercise.
(a) Show that a module $P$ is finitely generated if and only if the union of a totally ordered family of proper submodules of $P$ is a proper submodule.
(b) Show that $\text{Hom}_A(P,\bullet)$ preserves coproducts if and only if the union of every (countable) chain of proper submodules is a proper submodule.
(c) Show that the conditions in (a) and (b) are not equivalent. (Examples are not easy to find.)