  In this paper, we consider irreducible quasi-finite (or equivalently weakly
integrable) modules, with non-trivial action of the core, over the extended
affine Lie algebras (EALAs) whose centerless cores are multiloop algebras. The
centerless cores of all but one family of EALAs having nullity greater than 1
are known to admit such multiloop realizations. For any such (untwisted) EALA,
we show that the irreducible quasi-finite modules are either integrable with
the center of the underlying core acting trivially, or restricted generalized
highest weight (GHW) modules. We further prove that in the nullity 2 case,
these irreducible restricted GHW modules turn out to be highest weight type
modules, thereby classifying the irreducible quasi-finite modules over all such
EALAs. In particular, we obtain the classification of irreducible quasi-finite
modules over toroidal Lie algebras, minimal EALAs and toroidal EALAs of nullity
2. Along the way, we completely classify the irreducible weakly integrable
modules over affine Kac-Moody algebras (studied by Rao-Futorny [Trans. Amer.
Math. Soc. 2009] for non-zero level modules). Our results generalize the
well-known work of Chari [Invent. Math. 1986] and Chari-Pressley [Math. Ann.
1986] concerning the classification of irreducible integrable modules over
(nullity 1) affine Kac--Moody algebras.
