Scripture tells us that the original Temple built by Solomon was in the City of David. (First Kings 3:1) The Temple was built on the threshing floor of Ornan the Jebusite (Second Chronicles 3:1) which was purchased by King David. (Second Samuel 24:20-24) The area of the City of David also included the area known as the Ophel. (Second Chronicles 27:3) The Gihon Spring also was located at the City of David. (Second Chronicles 32:30, Second Chronicles 33:14) Solomon was crowned King of Israel at the Gihon Spring. (First Kings 1:38-39)
The common point of the threshing floor that King David bought, the Temple of Solomon and the Gihon Spring is the City of David, which is south of the traditional Temple Mount.
The Lord Jesus, while viewing the Temple, stated that not one stone will be left on another. They will all be thrown down. (Matthew 24:2) Today, the Jews pray at the Western Wall, which is many courses of stone high. In addition, a coin dated 20 A.D. has been found beneath the bottom stone course of the Western Wall, indicating that it was erected after the death of King Herod in 4 B.C.
The above facts are the basis for the current discussion about the actual location of the Temple of God. Tradition dies hard, and for centuries everyone has considered the traditional Temple Mount to be the site of the Temples of Solomon and Zerubbabel (which was later rebuilt by Herod.) However, there have been recent archaeological discoveries south of the traditional Temple Mount that indicate the Temple was located in the old City of David.
Opponents of the traditional Temple site believe that what we call the Temple Mount is actually the Roman fort built in Jerusalem. The size and shape of the traditional Temple Mount is very similar to other Roman forts built throughout the Roman Empire.
My personal opinion leans toward the City of David site south of the traditional Temple Mount, and the reason is Scripture. That is where the Bible indicates the Temple was located.
I bring this to your attention so that if you hear or read of this discussion, you will be aware of the basic fundamentals of the argument. There is much, much more. But I also have an interesting idea. Perhaps the Lord has allowed the world (and the Muslims) to think that the traditional Temple Mount was the location of His Temple. It is possible that if and when He wants the site at the City of David to be recognized and accepted, it will be done. And then the Jews can build their Temple on land that is not contested by the Arabs.
I do not know which site is correct, although I lean toward the City of David. But because I believe the time is near, we may soon see the revealing of the correct Temple site, wherever it is.