Large Language Models (LLM) and Vision Language Models (VLM) enable robots to ground natural language prompts into control actions to achieve tasks in an open world. However, when applied to a long-horizon collaborative task, this formulation results in excessive prompting for initiating or clarifying robot actions at every step of the task. We propose Language-driven Intention Tracking (LIT), leveraging LLMs and VLMs to model the human user's long-term behavior and to predict the next human intention to guide the robot for proactive collaboration. We demonstrate smooth coordination between a LIT-based collaborative robot and the human user in collaborative cooking tasks.