Linking (aligning) biomedical concepts across diverse data sources enables various integrative analyses, but it is challenging due to the discrepancies in concept naming conventions. Various strategies have been developed to overcome this challenge, such as those based on string-matching rules, manually crafted thesauri, and machine learning models. However, these methods are constrained by limited prior biomedical knowledge and can hardly generalize beyond the limited amounts of rules, thesauri, or training samples. Recently, large language models (LLMs) have exhibited impressive results in diverse biomedical NLP tasks due to their unprecedentedly rich prior knowledge and strong zero-shot prediction abilities. However, LLMs suffer from issues including high costs, limited context length, and unreliable predictions. In this research, we propose PromptLink, a novel biomedical concept linking framework that leverages LLMs. It first employs a biomedical-specialized pre-trained language model to generate candidate concepts that can fit in the LLM context windows. Then it utilizes an LLM to link concepts through two-stage prompts, where the first-stage prompt aims to elicit the biomedical prior knowledge from the LLM for the concept linking task and the second-stage prompt enforces the LLM to reflect on its own predictions to further enhance their reliability. Empirical results on the concept linking task between two EHR datasets and an external biomedical KG demonstrate the effectiveness of PromptLink. Furthermore, PromptLink is a generic framework without reliance on additional prior knowledge, context, or training data, making it well-suited for concept linking across various types of data sources. The source code is available at https://github.com/constantjxyz/PromptLink.