Large Language Models (LLMs) generate text token-by-token in discrete time, yet real-world communication, from therapy sessions to business negotiations, critically depends on continuous time constraints. Current LLM architectures and evaluation protocols rarely test for temporal awareness under real-time deadlines. We use simulated negotiations between paired agents under strict deadlines to investigate how LLMs adjust their behavior in time-sensitive settings. In a control condition, agents know only the global time limit. In a time-aware condition, they receive remaining-time updates at each turn. Deal closure rates are substantially higher (32\% vs. 4\% for GPT-5.1) and offer acceptances are sixfold higher in the time-aware condition than in the control, suggesting LLMs struggle to internally track elapsed time. However, the same LLMs achieve near-perfect deal closure rates ($\geq$95\%) under turn-based limits, revealing the failure is in temporal tracking rather than strategic reasoning. These effects replicate across negotiation scenarios and models, illustrating a systematic lack of LLM time awareness that will constrain LLM deployment in many time-sensitive applications.