In this paper, we study more general version of the paper [J. H. Lee, I. S. Kong, H. S. Kim and J. U. Jung, Generalized int-soft subsemigroups, Ann. Fuzzy Math. Inform. 8(6) (2014) 869-887]. We introduce the notion of int-soft semigroup with two thresholds ε and δ (briefly, (ε, δ)-int-soft semigroup) of a semigroup S, and investigate several related properties.