Perumal is inside as an antaryAmi, an in-dweller. There is no doubt that he is the Soul of our soul.
Notes:
According to Swami Desikan, as the Jivan is atomic, there is no such thing as inside of a Jivan. Hence there is no question of Permal being inside a soul. If Perumal is said to be an antaryAmi for a soul (Jivan), it implies that no part of a soul that is disassociated from Perumal, everything is attached to Him.