Large Language Models (LLMs) have recently demonstrated significant potential in the field of time series forecasting, offering impressive capabilities in handling complex temporal data. However, their robustness and reliability in real-world applications remain under-explored, particularly concerning their susceptibility to adversarial attacks. In this paper, we introduce a targeted adversarial attack framework for LLM-based time series forecasting. By employing both gradient-free and black-box optimization methods, we generate minimal yet highly effective perturbations that significantly degrade the forecasting accuracy across multiple datasets and LLM architectures. Our experiments, which include models like LLMTime with GPT-3.5, GPT-4, LLaMa, and Mistral, TimeGPT, and TimeLLM show that adversarial attacks lead to much more severe performance degradation than random noise, and demonstrate the broad effectiveness of our attacks across different LLMs. The results underscore the critical vulnerabilities of LLMs in time series forecasting, highlighting the need for robust defense mechanisms to ensure their reliable deployment in practical applications. The code repository can be found at https://github.com/JohnsonJiang1996/AdvAttack_LLM4TS.