Coupled nanomechanical resonators have recently attracted great attention for both practical applications and fundamental studies owing to their sensitive sympathetic oscillation dynamics. A challenge to the further development of this architecture is the coherent manipulation of the coupled oscillations. Here, we demonstrate strong dynamic coupling between two GaAs-based mechanical resonators by periodically modulating (pumping) the stress using a piezoelectric transducer. This strong coupling enables coherent transfer of phonon populations between the resonators, namely phonon Rabi oscillations. The nature of the dynamic coupling can also be tuned from a linear first-order interaction to a nonlinear higher-order process in which more than one pump phonon mediates the coherent oscillations (that is, multi-pump phonon mixing). This coherent manipulation is not only useful for controlling classical oscillations but can also be extended to the quantum regime, opening up the prospect of entangling two distinct macroscopic mechanical objects.