While Large Language Models (LLMs) excel at temporal reasoning tasks like event ordering and duration estimation, their ability to perceive the actual passage of time remains unexplored. We investigate whether LLMs perceive the passage of time and adapt their decision-making accordingly through three complementary experiments. First, we introduce the Token-Time Hypothesis, positing that LLMs can map discrete token counts to continuous wall-clock time, and validate this through a dialogue duration judgment task. Second, we demonstrate that LLMs could use this awareness to adapt their response length while maintaining accuracy when users express urgency in question answering tasks. Finally, we develop BombRush, an interactive navigation challenge that examines how LLMs modify behavior under progressive time pressure in dynamic environments. Our findings indicate that LLMs possess certain awareness of time passage, enabling them to bridge discrete linguistic tokens and continuous physical time, though this capability varies with model size and reasoning abilities. This work establishes a theoretical foundation for enhancing temporal awareness in LLMs for time-sensitive applications.