Accurate downlink channel state information (CSI) is one of the essential requirements for harnessing the potential advantages of frequency-division duplexing (FDD) massive multi-input multi-output (MIMO) systems. The current state-of-art in this vibrant research area include the use of deep learning to compress and feedback downlink CSI at the user equipments (UEs). These approaches focus mainly on achieving CSI feedback with high reconstruction performance and low complexity, but at the expense of inflexible compression rate (CR). High training overheads and limited storage capacity requirements are some of the challenges associated with the design of dynamic CR, which instantaneously adapt to propagation environment. This paper applies transfer learning (TL) to develop a multi-rate CSI compression and recovery neural network (TL-MRNet) with reduced training overheads. Simulation results are presented to validate the superiority of the proposed TL-MRNet over traditional methods in terms of normalized mean square error and cosine similarity.