The human brain has evolved specialised mechanisms to enable the rapid detection of threat cues, including emotional face expressions (e.g., fear and anger). However, contextual cues - such as gaze direction - influence the ability to recognise emotional expressions. For instance, anger paired with direct gaze, and fear paired with averted gaze are more accurately recognised compared to alternate conjunctions of these features. It is argued that this is because gaze direction conveys the relevance and locus of the threat to the observer. Here, we used continuous flash suppression (CFS) to assess whether the modulatory effect of gaze direction on emotional face processing occurs outside of conscious awareness. Previous research using CFS has demonstrated that fearful facial expressions are prioritised by the visual system and gain privileged access to awareness over other expressed emotions. We hypothesised that if the modulatory effects of gaze on emotional face processing occur also at this level, then the gaze-emotion conjunctions signalling self-relevant threat will reach awareness faster than those that do not. We report that fearful faces gain privileged access to awareness over angry faces, but that gaze direction does not modulate this effect. Thus, our findings suggest that previously reported effects of gaze direction on emotional face processing are likely to occur once the face is detected, where the self-relevance and locus of the threat can be consciously appraised.