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Deep clustering has shown its promising capability in joint representation learning and clustering via deep neural networks. Despite the significant progress, the existing deep clustering works mostly utilize some distribution-based clustering loss, lacking the ability to unify representation learning and multi-scale structure learning. To address this, this paper presents a new deep clustering approach termed image clustering with contrastive learning and multi-scale graph convolutional networks (IcicleGCN), which bridges the gap between convolutional neural network (CNN) and graph convolutional network (GCN) as well as the gap between contrastive learning and multi-scale structure learning for the deep clustering task. Our framework consists of four main modules, namely, the CNN-based backbone, the Instance Similarity Module (ISM), the Joint Cluster Structure Learning and Instance reconstruction Module (JC-SLIM), and the Multi-scale GCN module (M-GCN). Specifically, the backbone network with two weight-sharing views is utilized to learn the representations for the two augmented samples (from each image). The learned representations are then fed to ISM and JC-SLIM for joint instance-level and cluster-level contrastive learning, respectively, during which an auto-encoder in JC-SLIM is also pretrained to serve as a bridge to the M-GCN module. Further, to enforce multi-scale neighborhood structure learning, two streams of GCNs and the auto-encoder are simultaneously trained via (i) the layer-wise interaction with representation fusion and (ii) the joint self-adaptive learning. Experiments on multiple image datasets demonstrate the superior clustering performance of IcicleGCN over the state-of-the-art. The code is available at https://github.com/xuyuankun631/IcicleGCN.