While audio-visual speech models can yield superior performance and robustness compared to audio-only models, their development and adoption are hindered by the lack of labeled and unlabeled audio-visual data and the cost to deploy one model per modality. In this paper, we present u-HuBERT, a self-supervised pre-training framework that can leverage both multimodal and unimodal speech with a unified masked cluster prediction objective. By utilizing modality dropout during pre-training, we demonstrate that a single fine-tuned model can achieve performance on par or better than the state-of-the-art modality-specific models. Moreover, our model fine-tuned only on audio can perform well with audio-visual and visual speech input, achieving zero-shot modality generalization for multiple speech processing tasks. In particular, our single model yields 1.2%/1.4%/27.2% speech recognition word error rate on LRS3 with audio-visual/audio/visual input.