As indicated by the third author [Proc. Amer. Math. Soc. 146 (2018), pp. 4899-4903], there is a gap in the previous version of this paper by the first two authors [Three-dimensional noncompact κ-solutions that are Type I forward and backward, arXiv:1606.02698v1 (2016)]. We provide in this version an argument to fix the aforementioned gap. The main proposition, whose proof uses Perelman's techniques, is implied by Ding [Pacific J. Math. 240 (2009), pp. 289-308] and is covered by Hallgren [The nonexistence of noncompact Type-I ancient 3-d κ-solutions of Ricci flow with positive curvature, arXiv:1801.08643 (2018)] and the third author [Proc. Amer. Math. Soc. 146 (2018), pp. 4899-4903]. Our approach, however, is different from theirs. In addition, we prove a necessary and sufficient condition for a three-dimensional κ-solution to form a forward singularity. We hope that this condition is helpful in the classification of all three-dimensional κ-solutions. Up to now, the only main progress on such a classification, as conjectured by Perelman, is by Brendle [Invent. Math. 194 (2013), pp. 731-764].