There are four opinions where on Har Habayis the Beis Hamikdash was, they are:
1)dome of the rock is where the Kodesh Hakodoshim was.
2)dome of the rock is the place of the mizbeach.
3)Beis Hamikdash was in the south part of Har Habayis, in the empty space between the mosques.
4)Beis Hamikdash was in northern part of Har Habayis.
Each opinion can be sub-divided into a few opinions, arguing exactly the five-hundred amah square was, and the size of an amah.
(For a good summary of most of the opinions, as well as the problems with each one, see אוצר ירושלים והמקדש by Encyclopedia Talmudis, pages 325-345, and the diagrams in the back.)
1) The Dome of the Rock is the Place of the Kodesh Hakodoshim
This opinion holds that the rock in the dome of the rock is the Even Hashesiyah. This Opinion is found in many places, going back to the time of the Geonim, for example R' Binyamin from Toledo, R' Pesachyah from Regensburg, and the Radvaz (in שאלות ותשובות חלק ב', תשובה תרצ"א). This opinion itself is now divided in two, arguing whether Hurdus also expanded to the west, or only to the north and south.
R' Zalman Menachem Koren's opinion: R' Zalman Menachem Koren (חצרות בית ה', ירושלים תשל"ז) holds that Hurdus only expanded to the north and south, not to the west. He calculated an amah to be 57.4 centimeters. He says based on the Yerushalmi (Eiruvin 5:1) that the Beis Hamikdash was perfectly aligned with true east-west-north-south. He also holds that the Cheil had walls, and it is the platform that is now on Har Habayis. He also holds that the Ezras Nashim was not perfectly square, but the north wall was shorter than the south wall.
The problem with his opinion is, that the Ezras Nashim is not square, and the Antonia fortress is very big, but Yosifun (War of the Jews 5:5) writes that it was like a tower.
The problems with this opinion are, that Yerushalmi says that the Beis Hamikdash was perfectly aligned with east-west-north-south, and according to them it wasn't, and that Sha'ar Shushan was directly in front of Sha'ar Nicanor, and according to them it wasn't.
2)The Dome of the Rock is Where the Mizbeach Was
This opinion holds that the rock is the place of the mizbeach. It is held by Rabbi Shlomo Goren (הר הבית (משיב מלחמה, ד') ירושלים תשס"ה)
The problems with this opinion are,
3) The Southern Opinion
This opinion holds that the Beis Hamikdash was in the empty space between the two mosques. This opinion was held by many people, like R' Michael Braur (אבן שתיה או אבן הטועים, ירושלים תר"ץ), Yosef Rofeh, Ge'alyah Kornfeld, Tuvia Sagiv, R' Sholom Dov Steinberg (בית המקדש השלישי, ירושלים תרנ"ג), and R' Dov Levononi(המקדש בירושלים, ירושלים תשס"ד-The Temple In Jerusalem), with some small differences between these opinions. One of the proofs for this opinion is the Kaftor VaFerach (perek Vav) who says Sha'ar Shushan is in the southern third of the east wall, which if you say he is talking about the east wall of Har Habayis, is right in front of the empty space. (But you can also say that he is talking about the east wall of Yerushalayim, in which case you don't have any proof.)
Tuvia Sagiv's opinion
The problems with this opinion are,
4) The Northern Opinion
This opinion holds that the Beis Hamikdash was to the north of the dome of the rock, and that Sha'ar Harachamim (which is in the eastern wall of Har Habayis, north of the dome of the rock) is Sha'ar Shushan. This opinion was held by R' Emanuel Chay Riki, (in Aderes Eliyahu, Kuntres Mei Niddah, perek lamed-vav,) who based how big an amah is, on this gate. There is also, lehavdil, non-Jews who held this opinion, like Melchior De Vogue.
Asher Kaufman (The Temple Mount: where is the holy of holies, Jerusalem 2004) made a detailed description of the Har Habayis and Beis Hamikdash based on this opinion. According to him, the Kodesh Hakodoshim was where the dome of the spirits now stands, and the stone underneath it is the Even Hashesiyah. However, he holds that an amah is 43.7 centimeters, and that the Har Habayis and Beis Hamikdash are not square. He also holds that when the Tanna says Har Habayis is 500 amos by 500 amos, he means that it was 250,000 square amos, but not that it was a square 500 by 500, and that the the Beis Hamikdash is not included in this amount.
The big problem with his opinion is, that the Mishna says 500 long and 500 wide, and by Azarah 187 by 135, not how much square amos it was! (For more problems with his opinion, see אוצר ירושלים והמקדש, pages 338-339)
In the next post I will Im"h say a new way how it was, based on the Tavnis Heichal.
I have added a new video of the Lishkos on the northern side of the first Beis Hamikdash..
My name is Mendel Lewis.
Hashem said to Yechezkel, "Its reading in the Torah is as great as its building. Go and say it to them, and they will occupy themselves to read the form of it in the Torah. And in reward for its reading, that they occupy themselves to read about it, I count it for them as if they were occupied with the building of it. (Tanchuma tzav 14)