Radiation-pressure interactions between harmonic oscillators have enabled exquisite measurement precision and control, made possible by using strong sideband drives, enhancing the coupling rate while also linearizing the interaction. In this Letter, we demonstrate a strong intrinsic longitudinal coupling, a circuit quantum electrodynamics analog of the radiation-pressure interaction, between a transmon qubit and a linear microwave resonator. A red-detuned sideband drive results in an on-demand Jaynes-Cummings interaction with a high on-off ratio. We measure a longitudinal coupling rate an order of magnitude larger than all decay rates, placing the device in the strong coupling regime. The intrinsic longitudinal interaction demonstrated here will enable the development of high-connectivity quantum information processing hardware and the exploration of the gravitational decoherence of quantum objects.