Large language models (LLMs) have shown their potential in long-context understanding and mathematical reasoning. In this paper, we study the problem of using LLMs to detect tabular anomalies and show that pre-trained LLMs are zero-shot batch-level anomaly detectors. That is, without extra distribution-specific model fitting, they can discover hidden outliers in a batch of data, demonstrating their ability to identify low-density data regions. For LLMs that are not well aligned with anomaly detection and frequently output factual errors, we apply simple yet effective data-generating processes to simulate synthetic batch-level anomaly detection datasets and propose an end-to-end fine-tuning strategy to bring out the potential of LLMs in detecting real anomalies. Experiments on a large anomaly detection benchmark (ODDS) showcase i) GPT-4 has on-par performance with the state-of-the-art transductive learning-based anomaly detection methods and ii) the efficacy of our synthetic dataset and fine-tuning strategy in aligning LLMs to this task.