The two Jewish Temples were not located on the current location known as the Temple Mount. That location is where the Roman Fort Antonia was built.
Archaeology, history, and Jewish Rabbis make the case that the Temples were built adjacent to the City of David above the Gihon Spring.
Here is a link to Ernest Martin’s book The Temples that Jerusalem Forgot. I put a link for the Associates for Scriptural Knowledge in the links section on the right.
Here are some videos:
Andrew Gabriel Roth website here.
Daniel Silas
Hold the Line 
Hello Mr. Roth, I very appreciate the research on topics like the ancient handwritings of the Bible, like the quotes of the churchfathers about the Gospel in Hebrew, and the Aramaic language.
At Bibleschool they taught Marcus as source nr. 1, Q, but I always doubted that. And now you gave me the foundations of my doubt. Thank you.
A topic I disagree with you is the location of the Temple. I already supported the traditional location, but because you chose different, I had to study it again. I went through Josephus, but all his writing about the Temple location and the Tower of Antonia pointed toward the traditional location. Especially Wars Book V, 5, 1.
You found texts of Jewish scholars like the Rambam. May be they give a different light. Could you let me know where I can find those texts?
Yours
Jeroen Vonk
The Netherlands