In an airborne passive radar, multipath (MP) clutter, which is caused by MP signals contained in the contaminated reference signal, degrades the space-time adaptive processing (STAP) performance. The MP clutter suppression algorithm before STAP can mitigate the influence of impure reference signals. However, the performances of the existing MP clutter suppression methods deteriorate when the intrinsic clutter motion (ICM) exists because the sparse model of MP clutter is disturbed. To eliminate the impacts of ICM on MP clutter suppression, a joint optimization algorithm is developed for airborne passive radar. Firstly, the sparse model of MP clutter is modified by taking ICM fluctuation into account. Subsequently, the joint optimization function of the ICM fluctuation and MP clutter profile is derived. Finally, based on the local search technique, MP clutter is suppressed with ICM error calibration and off-grid effects mitigation. A range of simulations verify the reliability and superiority of the proposed method.