Some time-dependent reliability analysis methods use surrogate models to approximate the implicit limit state functions of complex systems. However, the performance of these methods is usually affected by the situations that the used models are not accurate and some samples have no significant contribution to the accuracy improvement. To construct a more suitable model for reliability analysis, this work proposes an active failure-pursuing Kriging modeling method to identify the most valuable samples for improving the accuracy of the predicted failure probability. On the one hand, a global predicted failure probability error index calculated through the real-time reliability result is proposed to pursue the sensitive sample and the corresponding local region that is most likely to maximize the improvement of the accuracy of the reliability result. A fault-tolerant scheme is further applied to ensure the accuracy of the failure-pursuing process. On the other hand, the correlation-based screening and space partition strategy is developed to describe the local regions and avoid the clustering of samples. In each iteration, the Kriging model is updated with the exploitation of new sample from the local regions around the sensitive samples. Additionally, an equivalent stochastic process transformation is developed to form a uniform high probability density sampling space. The results of three cases demonstrate the efficiency, accuracy and stability of the proposed method.