Both "aware" and "conscious" can be translated as "conscience" in French, but "conscious" and "aware" have different meanings (https://www.tandfonline.com/doi/abs/10.1080/09602010500484581).
If we consider that "conscience" is the French equivalent of "conscious" or "consciousness", what could be a (near) exact translation of "aware"/"awareness"?
EDIT (after having seen user @Dimitris comment and answer):
The article I pointed out highlights the differences in English between "conscious" and "aware" (mind/spirit/mental-state wise) which are often used interchangeably. The author describes awareness as a subset of consciousness, and "aware" doesn't seem to have a proper nuanced equivalent in French. As a native French speaker, the only translation I could think about is awake = éveillé. But I am not sure if it is correct.