Accurate channel state information (CSI) is necessary for frequency-division duplexing (FDD) massive multi-input multi-output (MIMO) systems. Existing deep learning-based CSI feedback methods, e.g., CSI sensing and recovery neural network (CsiNet), designed based on an autoencoder architecture, achieves higher feedback accuracy and reconstruction speed. However, this network needs to be retrained due to different communication scenarios and channel conditions, which is costly in practical deployment. To solve this problem, this paper proposes a deep learning-based modular adaptive multiple-rate (MAMR) compression CSI feedback framework. Extra padding modules are added at the base station to pad compressed CSI into different compression rates into the same dimensions, thereby realizing a general autoencoder performing variable-rate compression. Simulation results are given to confirm the effectiveness of the proposed method in terms of normalized mean square error.