Going off CodeNewbie's answer, I think the answer might be
I have always been there,
and will always be.
The present has always existed, and will continue to exist.
I'll be gone in a mere
moment, you'll see.
Your current point in time is the present, but in a moment that point in time will be the past.
You cannot remember
me to not be with you,
I don't actually understand this part, I'm having trouble parsing it, but probably something to do with anything you "remember" is in the past, not the present.
yet just at the moment,
I'm completely new.
The present moment is always a new moment.
You cannot escape me,
nor stay within me.
You will always be in the present, but you can never stay in the present moment.
Now could you please tell me,
do you recognize me?