Large language models (LLMs) have transformative potential in radiology, including textual summaries, diagnostic decision support, proofreading, and image analysis. However, the rapid increase in studies investigating these models, along with the lack of standardized LLM-specific reporting practices, affects reproducibility, reliability, and clinical applicability. To address this, reporting guidelines for LLM studies in radiology were developed using a two-step process. First, a systematic review of LLM studies in radiology was conducted across PubMed, IEEE Xplore, and the ACM Digital Library, covering publications between May 2023 and March 2024. Of 511 screened studies, 57 were included to identify relevant aspects for the guidelines. Then, in a Delphi process, 20 international experts developed the final list of items for inclusion. Items consented as relevant were summarized into a structured checklist containing 32 items across six key categories: general information and data input; prompting and fine-tuning; performance metrics; ethics and data transparency; implementation, risks, and limitations; and further/optional aspects. The final FLAIR (Framework for LLM Assessment in Radiology) checklist aims to standardize reporting of LLM studies in radiology, fostering transparency, reproducibility, comparability, and clinical applicability to enhance clinical translation and patient care.