Precipitation nowcasting is one of the main tasks of weather forecasting that aims to predict rainfall events accurately, even in low-rainfall regions. It has been observed that few studies have been devoted to predicting future radar echo images in a reasonable time using the deep learning approach. In this paper, we propose a novel approach, RainPredRNN, which is the combination of the UNet segmentation model and the PredRNN_v2 deep learning model for precipitation nowcasting with weather radar echo images. By leveraging the abilities of the contracting-expansive path of the UNet model, the number of calculated operations of the RainPredRNN model is significantly reduced. This result consequently offers the benefit of reducing the processing time of the overall model while maintaining reasonable errors in the predicted images. In order to validate the proposed model, we performed experiments on real reflectivity fields collected from the Phadin weather radar station, located at Dien Bien province in Vietnam. Some credible quality metrics, such as the mean absolute error (MAE), the structural similarity index measure (SSIM), and the critical success index (CSI), were used for analyzing the performance of the model. It has been certified that the proposed model has produced improved performance, about 0.43, 0.95, and 0.94 of MAE, SSIM, and CSI, respectively, with only 30% of training time compared to the other methods.