One of the key tasks in machine learning for tabular data is feature engineering. Although it is vital for improving the performance of models, it demands considerable human expertise and deep domain knowledge, making it labor-intensive endeavor. To address this issue, we propose a novel framework, \textbf{FeRG-LLM} (\textbf{Fe}ature engineering by \textbf{R}eason \textbf{G}eneration \textbf{L}arge \textbf{L}anguage \textbf{M}odels), a large language model designed to automatically perform feature engineering at an 8-billion-parameter scale. We have constructed two-stage conversational dialogues that enable language models to analyze machine learning tasks and discovering new features, exhibiting their Chain-of-Thought (CoT) capabilities. We use these dialogues to fine-tune Llama 3.1 8B model and integrate Direct Preference Optimization (DPO) to receive feedback improving quality of new features and the model's performance. Our experiments show that FeRG-LLM performs comparably to or better than Llama 3.1 70B on most datasets, while using fewer resources and achieving reduced inference time. It outperforms other studies in classification tasks and performs well in regression tasks. Moreover, since it does not rely on cloud-hosted LLMs like GPT-4 with extra API costs when generating features, it can be deployed locally, addressing security concerns.