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In this paper, for every $\epsilon\in \mathbb{Z}$, we introduce an extensionof the 2-toroidal Lie algebra by certain derivations. Based on the$\phi_\epsilon$-coordinated modules theory for vertex algebras, we give anexplicit realization of a class of irreducible highest weight modules for thisextended toroidal Lie algebra. When $\epsilon=1$, this affords a realization ofcertain irreducible modules for the toroidal extended affine Lie algebras firstconstructed by Billig.

Irreducible modules of toroidal Lie algebras arising from $\phi_\epsilon$-coordinated modules of vertex algebras