Telepresence refers to a set of tools that allows a person to be “present” in a distant environment, by a sufficiently realistic representation of it through a set of multimodal stimuli experienced by the distant devices via its sensors. Immersive Telepresence follows this trend and, thanks to the capabilities given by virtual reality devices, replicates distant sight and sound perception in a more “immersive” way. The use of coherent stereoscopic images displayed in a head mounted display, and natural control of a robotic head collocated with the orientation of the pilot head, help the pilot to feel “embodied” in the distant robotic platform. However even if the actual frameworks have shown increased awareness of the remote scene and enhanced interactivity, no work has currently addressed the challenge of gaze contingent controlled robotic eye in immersive teleoperation (ie. eye vergence) and its impact on scene awareness from the pilot and sense of presence from the remote interlocutors. Based on gaze driven technologies and analysis in related fields of study, we propose and evaluate a set of methods to quantify the impact of the proposed Stereo Gaze Contingent Steering (SGCS) of Robotic Eyes, notably on depth perception and near-field object perception.