I have heard different theories where the 2 Temples were located. Some say it was on or near the Dome of the Rock. Others say it was farther south, even possibly on Mt. Ophel between the al-Aqsa Mosque and Ir David, and the Dome of the Rock is really in middle of a former Herodian/Roman fortress (Antonia Fortress) which may have originally been built by the Hasmonean Kings.
Many Jews are under the impression that it was over the Rock, and the 'Sanhedrin' ruled this way.
Where exactly was the Beit Hamikdash located?