Multiple-input multiple-output (MIMO) radar three-dimensional (3D) imaging is widely applied in military and civil fields. However, MIMO is easily affected by wideband interference (WBI). To solve this problem, in this study, we propose a sparse recovery imaging method with WBI prediction based on the predictive recurrent neural network (PredRNN) and the tensor-based smooth L0 (TSL0) algorithm. Firstly, we extract the time-frequency (TF) feature of the historical measured WBI via the short-time Fourier transform (STFT) operation. In this way, we can use PredRNN to exploit the spatiotemporal correlation of the WBI in the TF domain to predict the TF feature of the WBI in the future. Then, we adaptively design the random sparse stepped frequency waveform by selecting non-overlapped frequencies with the WBI according to the predicted WBI TF feature. Finally, we apply the TSL0 algorithm to reconstruct the 3D high-resolution target image from the sparse signal cube. Simulation results show the high performance and robustness of the proposed imaging method in the presence of different WBIs.