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This paper is concerned with approximate solutions of multiobjective optimization problems whose order cone is not necessarily the nonnegative orthant. We introduce the concept of generalized ε-quasi-solution, that extends other well-known approximate efficiency notions of the literature, and we study the limit behavior of these solutions as approximations to the efficient and weak efficient sets. Moreover, we prove several existence results and a bound for the generalized ε-quasi-solution set under convexity assumptions and by using asymptotic analysis tools. Finally, we develop optimality conditions for a particular case of these kinds of ε-quasi-solutions in nonsmooth convex and nonconvex problems.