Anomaly detection in computational workflows is critical for ensuring system reliability and security. However, traditional rule-based methods struggle to detect novel anomalies. This paper leverages large language models (LLMs) for workflow anomaly detection by exploiting their ability to learn complex data patterns. Two approaches are investigated: 1) supervised fine-tuning (SFT), where pre-trained LLMs are fine-tuned on labeled data for sentence classification to identify anomalies, and 2) in-context learning (ICL) where prompts containing task descriptions and examples guide LLMs in few-shot anomaly detection without fine-tuning. The paper evaluates the performance, efficiency, generalization of SFT models, and explores zero-shot and few-shot ICL prompts and interpretability enhancement via chain-of-thought prompting. Experiments across multiple workflow datasets demonstrate the promising potential of LLMs for effective anomaly detection in complex executions.