We address the problem of audiovisual gaze control in the specific context of human-robot interaction, namely how controlled robot motions are combined with visual and acoustic observations in order to direct the robot head towards targets of interest. The paper has the following contributions: (i) a novel audiovisual fusion framework that is well suited for controlling the gaze of a robotic head; (ii) a reinforcement learning (RL) formulation for the gaze control problem, using a reward function based on the available temporal sequence of camera and microphone observations; and (iii) several deep architectures that allow to experiment with early and late fusion of audio and visual data. We introduce a simulated environment that enables us to learn the proposed deep RL model without the need of spending hours of tedious interaction. By thoroughly experimenting on a publicly available dataset and on a real robot, we provide empirical evidence that our method achieves state-of-the-art performance.