To be frank, I don't know my eyes move or not when I sleep. When I encounter your question, I look up some materials and get some information from my friends. Now I will give you my opinion. I hope it can disambiguate your question. Generally speaking, you are awake in your sleep even if your brain is shut down. Our sleep can be classified into two phases: the first few hours and the last hours of sleep before awakening. During the last hours of your sleep, your body needs little non-rem sleep time. In rem sleep, a person's eyes move back and forth rapidly. Thus, it's called "rapid eye movement" or rem sleep for short. Now, we get the answer that during the first few hours of our sleep, our eyes won't move, during the last hours of our sleep before awakening, our eyes begin to move. That's all!