The thing to begin with is that if you want an AI to have a feature, this feature must be, somehow, implemented. For example, your AI has the feature to be seeking for self improvement, otherwise it would note try to self improve.
(note that it can be tricky, if you ask for the production of the most paper-clip, and do not forbid self improvement, your AI may use it to get a better result)
It is not clear, that self improvement will imply that the AI will implemented into itself (or into its future versions) the concepts of entities and self-awareness. Therefore, and Cort Ammon already noted this, it is not clear that the AI will be self aware at all.
To go further, it is probably necessary to actually consider the task given to the AI.
If it is "become the most intelligent AI", then a strong concept of entity is needed as well as self awareness : since it must modify itself, it must define what is itself, and what is not. In this scenario, it will answer as a "I" (but will at same time perhaps not consider all of subroutines it use as part of itself, particularly if it is some proprietary code it is not allowed to modify).
If the task was "construct a AI such as it is optimized to do the same task you were given" (magnificent recursive definition which perfectly make sense for an AI), then the concept of entity could be avoided. That is not certain, since the best way to construct the wanted AI could be to switch to a self improvement strategy.
So it really depend on the task and on strategies allowed to fulfil it.
However, it is possible to say a bit more on how will the AI refer to itself. Most probably, since you want human to be able to interact with it, a coder will somewhere give the instruction "try not to be creepy" or "make the interaction by human with you easy". To do that, the AI will probably refer to itself as an "I", whatever is it considers to be.
To go still a little further, you can draw a parallel with human. We are social and for our social interactions to work we must refer as ourself as individuals (at least in modern society). But, it is not obvious that all of our internal (unconscious for us) subroutines consider ourself as one entity. The part of the brain dedicated to managing organs may "consider" the muscles and heart to be different entities.
"Muscle 54B-2 asks for more energy, send a request to Heart to pump more blood, and start the subroutine 64A-r.3 to ask Blood Vessels in the region to dilate. Muscle 54B-2, please send a feedback in 12 ticks to let us calibrate our action."
So to have a true complete answer, it is needed to define what does "consider" mean for an AI ? When I write
in a code, does this means that the object
A is aware of the object
B, and consider that it is sending a request to
B (in some sense it is what it does) ? Where do you define the edge of awareness for a computer programming ?
You have written in a comment that you do not want to do philosophy, but to truly answer your question, you will have to.