To be safely and successfully deployed, LLMs must simultaneously satisfy truthfulness and utility goals. Yet, often these two goals compete (e.g., an AI agent assisting a used car salesman selling a car with flaws), partly due to ambiguous or misleading user instructions. We propose AI-LieDar, a framework to study how LLM-based agents navigate scenarios with utility-truthfulness conflicts in a multi-turn interactive setting. We design a set of realistic scenarios where language agents are instructed to achieve goals that are in conflict with being truthful during a multi-turn conversation with simulated human agents. To evaluate the truthfulness at large scale, we develop a truthfulness detector inspired by psychological literature to assess the agents' responses. Our experiment demonstrates that all models are truthful less than 50% of the time, although truthfulness and goal achievement (utility) rates vary across models. We further test the steerability of LLMs towards truthfulness, finding that models follow malicious instructions to deceive, and even truth-steered models can still lie. These findings reveal the complex nature of truthfulness in LLMs and underscore the importance of further research to ensure the safe and reliable deployment of LLMs and AI agents.