Vibration signals from rolling bearings exhibit nonlinear and non-stationary characteristics. To improve the ability of remaining useful life (RUL) prediction methods in simultaneously capturing long-term dependency and local degradation information, this study proposes a convolutional white-box Transformer (CWTR) model for RUL prediction of rolling bearings. Firstly, a subspace attention mechanism integrating dilated causal convolution is designed to expand the receptive field of the attention mechanism and enhance the modeling ability of local dependency relationships in signals. Secondly, a multi-scale convolutional module is constructed to improve the interactive modeling ability of channel features under different time scales, allowing for finer extraction of local features at different degradation stages. Additionally, a Pearson correlation coefficient-based method is introduced to quantitatively assess the health status of bearings. Finally, an improved loss function is applied to optimize network training. Experiments are conducted on the real bearing dataset and the prediction results are compared with those of other prediction models. The root mean square error and mean absolute error are improved by 27.88% and 27.85% respectively, verifying the effectiveness of the CWTR model.