The site of the Temples (both Solomon’s and Herod’s) was in the City of David, south of what is wrongly called Temple Mount.
They were built in ZION… for location, see Wikipedia https://en.wikipedia.org/wiki/Zion
Here’s an example from Psalms from a plethora of Biblical evidence...
“May He send you help from the sanctuary,
And strengthen you out of Zion” (Psalm 20:2)
This is Hebrew poetic parallelism—repeating the idea twice in different form. It's self-evident that ‘the sanctuary’ (holy place or temple) and Zion are one and the same location. Zion was (and is) within the old City of David and is the true site of Jerusalem’s temples. Buy THE PLACE today to get the biblical history in fascinating narrative form https://wipfandstock.com/the-place.html