Fault localization (FL) targets identifying bug locations within a software system, which can enhance debugging efficiency and improve software quality. Due to the impressive code comprehension ability of Large Language Models (LLMs), a few studies have proposed to leverage LLMs to locate bugs, i.e., LLM-based FL, and demonstrated promising performance. However, first, these methods are limited in flexibility. They rely on bug-triggering test cases to perform FL and cannot make use of other available bug-related information, e.g., bug reports. Second, they are built upon proprietary LLMs, which are, although powerful, confronted with risks in data privacy. To address these limitations, we propose a novel LLM-based FL framework named FlexFL, which can flexibly leverage different types of bug-related information and effectively work with open-source LLMs. FlexFL is composed of two stages. In the first stage, FlexFL reduces the search space of buggy code using state-of-the-art FL techniques of different families and provides a candidate list of bug-related methods. In the second stage, FlexFL leverages LLMs to delve deeper to double-check the code snippets of methods suggested by the first stage and refine fault localization results. In each stage, FlexFL constructs agents based on open-source LLMs, which share the same pipeline that does not postulate any type of bug-related information and can interact with function calls without the out-of-the-box capability. Extensive experimental results on Defects4J demonstrate that FlexFL outperforms the baselines and can work with different open-source LLMs. Specifically, FlexFL with a lightweight open-source LLM Llama3-8B can locate 42 and 63 more bugs than two state-of-the-art LLM-based FL approaches AutoFL and AgentFL that both use GPT-3.5. In addition, FlexFL can localize 93 bugs that cannot be localized by non-LLM-based FL techniques at the top 1. Furthermore, to mitigate potential data contamination, we conduct experiments on a dataset which Llama3-8B has not seen before, and the evaluation results show that FlexFL can also achieve good performance.