In this paper, we investigate a dielectrically chiral-core fiber for the generation of orbital angular momentum (OAM) modes. We observe that the presence of chirality induces a modal index split between the same order OAM modes with opposite topological charges and arbitrary phase front rotation directions, which are originally degenerate in achiral fibers. This split indicates the existence of circular birefringence (CB) associated with a dielectric chirality. The modal cutoff splits result in a single polarization property without the possibility of coupling between the same order modes that correspond to a single +l or -l OAM mode guiding. However, neither of the two fundamental modes guided in the chiral fiber exhibits a cutoff, despite having different modal indices due to chirality. Upon fraction of modal power cutoff, the high-order modes in the core region display different cutoff fractions of modal power between the same order modes, while the fundamental mode has no cutoff fraction of modal power. Owing to the CB and different fractions of modal power in the core for different handed OAM modes, dielectrically chiral fibers have potential applications in chiral sensing, circular polarization-dependent OAM mode filters, and new kind of OAM mode generators.