Abstract Nonlinear normal modes (NNMs) are obtained for spacecraft relative motion in the case of bounded Keplerian reference orbits. For this purpose, the Lyapunov-Floquet transformation, invariant manifold-based order reduction, and time-dependent normal forms are employed on the third order extension of the Tschauner-Hempel equations to obtain the NNM manifolds and the reduced order dynamics on the NNMs. This approach extends the modal interpretation of relative orbital motion from the Hill-Clohessy-Wiltshire equations to the regime of elliptic reference orbits and large separation distances in which second and third order nonlinearities are non-negligible.