Large Language Models (LLMs), like ChatGPT, have gained widespread popularity and usage in various software engineering tasks, including refactoring, testing, code review, and program comprehension. Despite recent studies delving into refactoring documentation in commit messages, issues, and code review, little is known about how developers articulate their refactoring needs when interacting with ChatGPT. In this paper, our goal is to explore conversations between developers and ChatGPT related to refactoring to better understand how developers identify areas for improvement in code and how ChatGPT addresses developers' needs. Our approach relies on text mining refactoring-related conversations from 17,913 ChatGPT prompts and responses, and investigating developers' explicit refactoring intention. Our results reveal that (1) developer-ChatGPT conversations commonly involve generic and specific terms/phrases; (2) developers often make generic refactoring requests, while ChatGPT typically includes the refactoring intention; and (3) various learning settings when prompting ChatGPT in the context of refactoring. We envision that our findings contribute to a broader understanding of the collaboration between developers and AI models, in the context of code refactoring, with implications for model improvement, tool development, and best practices in software engineering.