A report in Science, Movement Intention after Parietal Cortex Stimulation in Humans, by M. Desnurget and others, has the following summary:
Parietal and premotor cortex regions are serious contenders for bringing motor intentions and motor responses into awareness. We used electrical stimulation in seven patients undergoing awake brain surgery. Stimulating the right inferior parietal regions triggered a strong intention and desire to move the contralateral hand, arm, or foot, whereas stimulating the left inferior parietal region provoked the intention to move the lips and to talk. When stimulation intensity was increased in parietal areas, participants believed they had really performed these movements, although no electromyographic activity was detected. Stimulation of the premotor region triggered overt mouth and contralateral limb movements. Yet, patients firmly denied that they had moved. Conscious intention and motor awareness thus arise from increased parietal activity before movement execution.
So the parietal region is involved in the conscious experience of intention and desire to move (ie the will to move) and the conscious experience of having moved. It is not involved in the movement itself. On the other hand, the premotor region is involved in the movement’s execution but not the the conscious experience of the movement.
The key here may be that the construction of conscious experience is a projection in time of that will be happening later, at the time of the experience. The construction process would therefore need to have access to motor programs that are being created (or even considered) so as to predict and project the sensory effect of the action before it has occurred.