Federated multi-view clustering is an emerging machine learning paradigm that groups the data with each view distributed on an isolated client while preserving their privacies. Although recent researches have proposed a few feasible solutions, they are severely limited by two drawbacks. In specific, the clients are required to share their data representations at each iteration of model training, leading to heavy communication overhead. On the other hand, existing researches handle large-scale data by employing the matrix factorization and neural network encoding techniques, failing to utilize their similarity information sufficiently. To address these issues, we propose a communication-efficient federated multi-view clustering framework by approximating the data representation with pseudo-label and centroid matrix, where the latter two are shared in model training. Meanwhile, the framework is instanced by incorporating linear kernel function to consider the data pairwise similarities. Note that, corresponding linear kernels are not required to compute explicitly, making the resultant method able to be optimized in linear complexity to the number of samples. Nevertheless, the proposed method is evaluated on benchmark datasets. It not only achieves inspiring results (26.84% accuracy improvement on average, 2.9$_\times$×-2153$_\times$× computation speedup and 98.4% communication overhead reduction at most) compared with existing federated multi-view clustering methods, but also outperforms centralized multi-view clustering approaches on performance and computation efficiency.