Large Language Models (LLMs) are increasingly used to support intelligent agents in knowledge-intensive tasks. Yet, their application to business process discovery remains underexplored, especially in multi-agent settings requiring coordinated, multi-turn dialogue. This paper introduces an LLM-based Multi-Agent Systems (MAS) framework that uncovers business processes through structured conversational interactions. Using the Gaia methodology, we designed four MAS types (Monolithic, Duo, Manager, and Team) with varying levels of agent role specialization and coordination complexity. A custom, model-agnostic infrastructure is used to support the choreography-based collaboration of LLM-based agents. We evaluated our LLM-based MAS across three realistic business processes and three LLMs (GeminiPro 2.5, Mistral Large 2, DeepSeek-V3). GeminiPro consistently outperformed other models, reaching 100% accuracy in some configuration-process pairs, while Mistral degraded sharply under complex setups. Although modular MAS designs enhanced prompt efficiency, they also introduced coordination overhead and failure risks. These findings demonstrate the feasibility of conversational MAS for process discovery and highlight the importance of aligning MAS system architecture with LLM model capabilities.