Robots operating in human-centric environments require the integration of visual grounding and grasping capabilities to effectively manipulate objects based on user instructions. This work focuses on the task of referring grasp synthesis, which predicts a grasp pose for an object referred through natural language in cluttered scenes. Existing approaches often employ multi-stage pipelines that first segment the referred object and then propose a suitable grasp, and are evaluated in private datasets or simulators that do not capture the complexity of natural indoor scenes. To address these limitations, we develop a challenging benchmark based on cluttered indoor scenes from OCID dataset, for which we generate referring expressions and connect them with 4-DoF grasp poses. Further, we propose a novel end-to-end model (CROG) that leverages the visual grounding capabilities of CLIP to learn grasp synthesis directly from image-text pairs. Our results show that vanilla integration of CLIP with pretrained models transfers poorly in our challenging benchmark, while CROG achieves significant improvements both in terms of grounding and grasping. Extensive robot experiments in both simulation and hardware demonstrate the effectiveness of our approach in challenging interactive object grasping scenarios that include clutter.