The existing neural architecture search (NAS) methods usually restrict the search space to the pre-defined types of block for a fixed macro-architecture. However, this strategy will limit the search space and affect architecture flexibility if block proposal search (BPS) is not considered for NAS. As a result, block structure search is the bottleneck in many previous NAS works. In this work, we propose a new evolutionary algorithm referred to as latency EvoNAS (LEvoNAS) for block structure search, and also incorporate it to the NAS framework by developing a novel two-stage framework referred to as Block Proposal NAS (BP-NAS). Comprehensive experimental results on two computer vision tasks demonstrate the superiority of our newly proposed approach over the state-of-the-art lightweight methods. For the classification task on the ImageNet dataset, our BPN-A is better than 1.0-MobileNetV2 with similar latency, and our BPN-B saves 23.7% latency when compared with 1.4-MobileNetV2 with higher top-1 accuracy. Furthermore, for the object detection task on the COCO dataset, our method achieves significant performance improvement than MobileNetV2, which demonstrates the generalization capability of our newly proposed framework.