Understanding probabilistic dependencies among variables is central to analyzing complex systems. Traditional structure learning methods often require extensive observational data or are limited by manual, error-prone incorporation of expert knowledge. Recent studies have explored using large language models (LLMs) for structure learning, but most treat LLMs as auxiliary tools for pre-processing or post-processing, leaving the core learning process data-driven. In this work, we introduce a unified framework for Bayesian network structure discovery that places LLMs at the center, supporting both data-free and data-aware settings. In the data-free regime, we introduce \textbf{PromptBN}, which leverages LLM reasoning over variable metadata to generate a complete directed acyclic graph (DAG) in a single call. PromptBN effectively enforces global consistency and acyclicity through dual validation, achieving constant $\mathcal{O}(1)$ query complexity. When observational data are available, we introduce \textbf{ReActBN} to further refine the initial graph. ReActBN combines statistical evidence with LLM by integrating a novel ReAct-style reasoning with configurable structure scores (e.g., Bayesian Information Criterion). Experiments demonstrate that our method outperforms prior data-only, LLM-only, and hybrid baselines, particularly in low- or no-data regimes and on out-of-distribution datasets. Code is available at https://github.com/sherryzyh/llmbn.