Benefiting from diverse instruction datasets, contemporary Large Language Models (LLMs) perform effectively as AI assistants in collaborating with humans. However, LLMs still struggle to generate natural and colloquial responses in real-world applications such as chatbots and psychological counseling that require more human-like interactions. To address these limitations, we introduce NICO, a Natural Interactive COnversation dataset in Chinese. We first use GPT-4-turbo to generate dialogue drafts and make them cover 20 daily-life topics and 5 types of social interactions. Then, we hire workers to revise these dialogues to ensure that they are free of grammatical errors and unnatural utterances. We define two dialogue-level natural conversation tasks and two sentence-level tasks for identifying and rewriting unnatural sentences. Multiple open-source and closed-source LLMs are tested and analyzed in detail. The experimental results highlight the challenge of the tasks and demonstrate how NICO can help foster the natural dialogue capabilities of LLMs. The dataset will be released.