In this paper, we propose a novel bidirectional multiscale feature aggregation (BMFA) network with attentional fusion modules for text-independent speaker verification. The feature maps from different stages of the backbone network are iteratively combined and refined in both a bottom-up and top-down manner. Furthermore, instead of simple concatenation or element-wise addition of feature maps from different stages, an attentional fusion module is designed to compute the fusion weights. Experiments are conducted on the NIST SRE16 and VoxCeleb1 datasets. The experimental results demonstrate the effectiveness of the bidirectional aggregation strategy and show that the proposed attentional fusion module can further improve the performance.