A nonlinear analysis of the inviscid stability of a cylindrical vortex enclosing a central jet of light or dense fluid is presented. A uniformly valid second‐order expansion is obtained using the method of multiple time scales. For a stagnant core surrounded by a potential vortex, the axisymmetric mode remains stable to various scales of disturbances independent of core density. For asymmetric modes, the nonlinear effects make the stagnant core vortex sheet more unstable in second order for a given value of the core density. A central jet of any finite density always has a destabilizing effect on the jet‐vortex system undergoing any mode of deformation. A jet of any velocity tends to increase the unstable growth rates with increasing order of perturbations. The surface tension has a stabilizing effect with increasing order of perturbations. The frequency of oscillation becomes amplitude dependent in various orders of perturbations. The cylindrical surface departs significantly from its initially perturbed sinusoidal form and assumes an asinusoidal form.