Radiologists rely on eye movements to navigate and interpret medical images. A trained radiologist possesses knowledge about the potential diseases that may be present in the images and, when searching, follows a mental checklist to locate them using their gaze. This is a key observation, yet existing models fail to capture the underlying intent behind each fixation. In this paper, we introduce a deep learning-based approach, RadGazeIntent, designed to model this behavior: having an intention to find something and actively searching for it. Our transformer-based architecture processes both the temporal and spatial dimensions of gaze data, transforming fine-grained fixation features into coarse, meaningful representations of diagnostic intent to interpret radiologists' goals. To capture the nuances of radiologists' varied intention-driven behaviors, we process existing medical eye-tracking datasets to create three intention-labeled subsets: RadSeq (Systematic Sequential Search), RadExplore (Uncertainty-driven Exploration), and RadHybrid (Hybrid Pattern). Experimental results demonstrate RadGazeIntent's ability to predict which findings radiologists are examining at specific moments, outperforming baseline methods across all intention-labeled datasets.