A large-axial-wavenumber asymptotic analysis of inviscid normal modes in an axisymmetric vortex with a weak axial flow is performed in this work. Using a WKBJ approach, general conditions for the existence of regular neutral modes are obtained. Dispersion relations are derived for neutral modes confined in the vortex core (‘core modes’) or in a ring (‘ring modes’). Results are applied to a vortex with Gaussian vorticity and axial velocity profiles, and a good agreement with numerical results is observed for almost all values of k. The theory is also extended to deal with singular modes possessing a critical point singularity. We demonstrate that the characteristics for vanishing viscosity of viscous damped normal modes can also be obtained. Known viscous damped eigenfrequencies for the Gaussian vortex without axial flow are, in particular, shown to be predicted well by our estimates. The theory is also shown to provide explanations for a few of their peculiar properties.