Cross-modal correlation provides an inherent supervision for video unsupervised representation learning. Existing methods focus on distinguishing different video clips by visual and audio representations. We human visual perception could attend to regions where sounds are made, and our auditory perception could also ground their frequencies of sounding objects, which we call bidirectional local correspondence. Such supervision is intuitive but not well explored in the contrastive learning framework. This paper introduces a pretext task, Cross-Modal Attention Consistency (CMAC), for exploring the bidirectional local correspondence property. The CMAC approach aims to align the regional attention generated purely from the visual signal with the target attention generated under the guidance of acoustic signal, and do a similar alignment for frequency grounding on the acoustic attention. Accompanied by a remoulded cross-modal contrastive loss where we consider additional within-modal interactions, the CMAC approach works effectively for enforcing the bidirectional alignment. Extensive experiments on six downstream benchmarks demonstrate that CMAC can improve the state-of-the-art performance on both visual and audio modalities.