In this paper, we propose a speaker verification method by an Attentive Multi-scale Convolutional Recurrent Network (AMCRN). The proposed AMCRN can acquire both local spatial information and global sequential information from the input speech recordings. In the proposed method, logarithm Mel spectrum is extracted from each speech recording and then fed to the proposed AMCRN for learning speaker embedding. Afterwards, the learned speaker embedding is fed to the back-end classifier (such as cosine similarity metric) for scoring in the testing stage. The proposed method is compared with state-of-the-art methods for speaker verification. Experimental data are three public datasets that are selected from two large-scale speech corpora (VoxCeleb1 and VoxCeleb2). Experimental results show that our method exceeds baseline methods in terms of equal error rate and minimal detection cost function, and has advantages over most of baseline methods in terms of computational complexity and memory requirement. In addition, our method generalizes well across truncated speech segments with different durations, and the speaker embedding learned by the proposed AMCRN has stronger generalization ability across two back-end classifiers.