In recent years, various software supply chain (SSC) attacks have posed significant risks to the global community. Severe consequences may arise if developers integrate insecure code snippets that are vulnerable to SSC attacks into their products. Particularly, code generation techniques, such as large language models (LLMs), have been widely utilized in the developer community. However, LLMs are known to suffer from inherent issues when generating code, including fabrication, misinformation, and reliance on outdated training data, all of which can result in serious software supply chain threats. In this paper, we investigate the security threats to the SSC that arise from these inherent issues. We examine three categories of threats, including eleven potential SSC-related threats, related to external components in source code, and continuous integration configuration files. We find some threats in LLM-generated code could enable attackers to hijack software and workflows, while some others might cause potential hidden threats that compromise the security of the software over time. To understand these security impacts and severity, we design a tool, SSCGuard, to generate 439,138 prompts based on SSC-related questions collected online, and analyze the responses of four popular LLMs from GPT and Llama. Our results show that all identified SSC-related threats persistently exist. To mitigate these risks, we propose a novel prompt-based defense mechanism, namely Chain-of-Confirmation, to reduce fabrication, and a middleware-based defense that informs users of various SSC threats.