  In this paper, we study nullity-2 toroidal extended affine Lie algebras in
the context of
  vertex algebras and their $\phi$-coordinated modules. Among the main results,
we introduce a variant of
  toroidal extended affine Lie algebras, associate vertex algebras to the
variant Lie algebras, and
  establish a canonical connection between modules for toroidal extended affine
Lie algebras and
  $\phi$-coordinated modules for these vertex algebras. Furthermore, by
employing some results of Billig,
  we obtain an explicit realization of irreducible modules for the variant Lie
algebras.
