Signal inpainting is the task of restoring degraded or missing samples in a signal. In this paper we address signal inpainting when Fourier magnitudes are observed. We propose a mathematical formulation of the problem that highlights its connection with phase retrieval, and we introduce two methods for solving it. First, we derive an alternating minimization scheme, which shares similarities with the Gerchberg-Saxton algorithm, a classical phase retrieval method. Second, we propose a convex relaxation of the problem, which is inspired by recent approaches that reformulate phase retrieval into a semidefinite program. We assess the potential of these methods for the task of inpainting gaps in speech signals. Our methods exhibit both a high probability of recovering the original signals and robustness to magnitude noise.