Can someone be aware while they are asleep?

It depends on what you mean by aware.

If you mean aware of the world around them, yes it, happens all the time, your senses, especially hearing, do not shut off when you fall asleep, but most of the time, what you hear will be incorporated into your dream.

If you mean aware that they are dreaming, or able to affect things in their dream, again, it is possible, Its called lucid dreaming.