If I understand that right, you are asking if dreams happen while we are asleep, rather than an illusion of what was happening in our brains which occurs only on waking.
If my take on you question is right then studies have shown that periods of rapid eye movement (REM) coincide with dreaming and dreams are only remembered if the person wakes during REM sleep.
Since waking up and not remembering any dream is common then it is unlikely to be a response to the waking process.
Hope that helps.
I am interested in dreams, but only wake remembering one every few months