There is Mahayana text which is believed to be taught by Maitreya and written down by Asanga, Madhyanta-vibhaga-karika. In this text, Maitreya introduces something he calls "false imagination" - which is a mode of cognition that involves imputing subject/object duality onto stuff.
The text is much deeper than my ability to explain it here, so I advise you to study it for yourself, but it does discuss this exact topic in great details. The simple answer is, yes of course it is possible as this mode of cognition is (a part of, a result of) Enlightenment.
In my limited understanding, the way of thinking that delineates individual objects, evaluates each object relatively to a frame of reference, and forms an attitude towards each object - is exactly what leads to emergence of self-sense. Once this way of thinking is overthrown, whether through analytical study followed by meditation, or through yogic exercises, or through wrestling with koans, or through selfless practice of Bodhisattva, or through direct cultivation of prajna-paramita etc. - once we no longer think in terms of entities and don't build attitudes to them - the sense of self is left with no food and begins to fade. And vice-versa.
In one interpretation, by Mario D'Amato in his translation of the text, this mode of cognition is completely free from semiotic interpretation of "signs" into "what they imply", and is instead a pure cognition of everything-as-it-is.
This seems close to my personal experience, when, as my practice progressed, I noticed that my tendency to "read-in" or impute my assumptions onto things and people have significantly reduced in direct proportion to diminishing sense of self. More and more I see things as they are, in all their unknown-to-me-depth and multifaceted ambiguity, instead of trying to assume I know them in a black-and-white way. This is directly connected to the absence of self, because self is a collection of reified assumptions.
At the end, the state of awareness without self seems similar to a state of mind when you drive car on a very familiar road. It's not like you are not aware of what you see - you are, otherwise you would have crashed or lost your way. But it is like, the road is so familiar that it creates no tension that would require some hard form of deliberate attention. Although the attention is definitely there. Similarly, when there is no self, it seems like awareness is not affected, but the sense of tension brought by the self is gone.
In one sense, it is somewhat (but not quite) like awareness of a child - meaning, there is so much harmony with the environment, that the sense of separate self is not required. It is a more harmonious kind of awareness, there is no conflict in it, because the conflict is self.