For a service robot, it is crucial to perceive as early as possible that an approaching person intends to interact: in this case, it can proactively enact friendly behaviors that lead to an improved user experience. We solve this perception task with a sequence-to-sequence classifier of a potential user intention to interact, which can be trained in a self-supervised way. Our main contribution is a study of the benefit of features representing the person's gaze in this context. Extensive experiments on a novel dataset show that the inclusion of gaze cues significantly improves the classifier performance (AUROC increases from 84.5% to 91.2%); the distance at which an accurate classification can be achieved improves from 2.4 m to 3.2 m. We also quantify the system's ability to adapt to new environments without external supervision. Qualitative experiments show practical applications with a waiter robot.