Conventional Task and Motion Planning (TAMP) approaches rely on manually designed interfaces connecting symbolic task planning with continuous motion generation. These domain-specific and labor-intensive modules are limited in addressing emerging tasks in real-world settings. Here, we present LLM3, a novel Large Language Model (LLM)-based TAMP framework featuring a domain-independent interface. Specifically, we leverage the powerful reasoning and planning capabilities of pre-trained LLMs to propose symbolic action sequences and select continuous action parameters for motion planning. Crucially, LLM3 incorporates motion planning feedback through prompting, allowing the LLM to iteratively refine its proposals by reasoning about motion failure. Consequently, LLM3 interfaces between task planning and motion planning, alleviating the intricate design process of handling domain specific messages between them. Through a series of simulations in a box-packing domain, we quantitatively demonstrate the effectiveness of LLM3 in solving TAMP problems and the efficiency in selecting action parameters. Ablation studies underscore the significant contribution of motion failure reasoning to the success of LLM3. Furthermore, we conduct qualitative experiments on a physical manipulator, demonstrating the practical applicability of our approach in real-world settings.