Consciousness, as described in the experimental literature, is a multi-faceted phenomenon, that impinges on other well-studied concepts such as attention and control. Do consciousness and attention refer to different aspects of the same core phenomenon, or do they correspond to distinct functions? One possibility to address this question is to examine the neural mechanisms underlying consciousness and attention. If consciousness and attention pertain to the same concept, they should rely on shared neural mechanisms. Conversely, if their underlying mechanisms are distinct, then consciousness and attention should be considered as distinct entities. This paper therefore reviews neurophysiological facts arguing in favor or against a tight relationship between consciousness and attention. Three neural mechanisms that have been associated with both attention and consciousness are examined (neural amplification, involvement of the fronto-parietal network, and oscillatory synchrony), to conclude that the commonalities between attention and consciousness at the neural level may have been overestimated. Last but not least, experiments in which both attention and consciousness were probed at the neural level point toward a dissociation between the two concepts. It therefore appears from this review that consciousness and attention rely on distinct neural properties, although they can interact at the behavioral level. It is proposed that a "cumulative influence model," in which attention and consciousness correspond to distinct neural mechanisms feeding a single decisional process leading to behavior, fits best with available neural and behavioral data. In this view, consciousness should not be considered as a top-level executive function but should rather be defined by its experiential properties.