Using the Landau-Ginzburg-Devonshire phenomenological approach, we perform finite element modeling of the electric polarization, electric field, and elastic stresses and strains in a core-shell nanoparticle, where the ferroelectric core has a shape of prolate cylinder. Calculations reveal the quadrupolar-type diffuse domain structure consisting of two oppositely oriented diffuse axial domains located near the cylinder ends, which are separated by a region with a zero-axial polarization; we have termed this “flexon” to underline the flexoelectric nature of its axial polarization. Analytical calculations and FEM results have proven that a change of the flexoelectric coefficient sign leads to a reorientation of the flexon axial polarization; as well as an anisotropy of the flexoelectric coupling critically influences the flexon formation and related domain morphology. The flexon polarization forms a drop-shaped region near the ends of the cylinder, with a distinct chiral structure that is determined by the sign of the flexoelectric coupling constant. Its rounded shape, combined with its distinct chiral properties and the localization nature near the surface are reminiscent of a those of Chiral Bobber structures in magnetism. In the azimuthal plane, the flexon displays the polarization state of a meron. We show that this new type of chiral polarization structure is stabilized by an anisotropic flexoelectric coupling. It is important to note that the Lifshitz invariant describing the flexoelectric effect, which couples the electric polarization and elastic strain gradients, plays a determining role in the stabilization of these chiral states. It thereby provides an energetic interaction that, similar to the recently predicted ferroelectric Dyzaloshinskii-Moryia interaction, can lead to the formation of chiral polarization states, and, by extension, ferroelectric skyrmions.