Language model (LM) based audio generation frameworks, e.g., AudioLM, have recently achieved new state-of-the-art performance in zero-shot audio generation. In this paper, we explore the feasibility of LMs for zero-shot voice conversion. An intuitive approach is to follow AudioLM - Tokenizing speech into semantic and acoustic tokens respectively by HuBERT and SoundStream, and converting source semantic tokens to target acoustic tokens conditioned on acoustic tokens of the target speaker. However, such an approach encounters several issues: 1) the linguistic content contained in semantic tokens may get dispersed during multi-layer modeling while the lengthy speech input in the voice conversion task makes contextual learning even harder; 2) the semantic tokens still contain speaker-related information, which may be leaked to the target speech, lowering the target speaker similarity; 3) the generation diversity in the sampling of the LM can lead to unexpected outcomes during inference, leading to unnatural pronunciation and speech quality degradation. To mitigate these problems, we propose LM-VC, a two-stage language modeling approach that generates coarse acoustic tokens for recovering the source linguistic content and target speaker's timbre, and then reconstructs the fine for acoustic details as converted speech. Specifically, to enhance content preservation and facilitates better disentanglement, a masked prefix LM with a mask prediction strategy is used for coarse acoustic modeling. This model is encouraged to recover the masked content from the surrounding context and generate target speech based on the target speaker's utterance and corrupted semantic tokens. Besides, to further alleviate the sampling error in the generation, an external LM, which employs window attention to capture the local acoustic relations, is introduced to participate in the coarse acoustic modeling.