The Ashkenazi Chief Rabbi of Israel David Lau stated the third Jewish Temple could be built on the Temple Mount without removing Muslim places of worship. He says there is plenty of room for all to worship.
“Jews, Christians, Muslims, everyone,” stated Rabbi Lau.
As of June 2016, Yehuda Glick is now a member of the Israeli Knesset. This is the same Yehuda Glick who took multiple gun shots from an intended assassin and survived. He is the same rabbi who was the head of the Temple Institute in Jerusalem. He believes Jews should be able to pray on the Temple Mount. Previously, Glick has stated a vision of three faiths (Islam, Christianity and Judaism) sharing the Temple Mount. This idea is gaining traction among Jews in Israel. Some believe it could be the key to a future solution to the Arab-Israeli conflict.
http://www.timesofisrael.com/israel-chief-rabbi-urges-rebuilding-jerusalem-temple/
There are multiple places where the Temple could be built on the Mount. The main focus appears to be in the olive grove on the north end of the Temple Mount. This location is particularly interesting. Straight east is the Golden Gate or the Eastern Gate. This is the Messiah’s gate. It has been closed for centuries.
Ezekiel 44:1-3 Then he brought me back to the outer gate of the sanctuary, which faces east. And it was shut. And the LORD said to me, “This gate shall remain shut; it shall not be opened, and no one shall enter by it, for the LORD, the God of Israel, has entered by it. Therefore it shall remain shut. Only the prince may sit in it to eat bread before the LORD. He shall enter by way of the vestibule of the gate, and shall go out by the same way.”
This location would also align with Revelation 11:2 Leave out the court which is outside the temple and do not measure it, for it has been given to the nations; and they will tread under foot the holy city for forty-two months.
A temple constructed in the north end of the Temple Mount would allow for the Dome of the Rock and Al-Aqsa Mosque to be in the outer courtyard of the Jewish Temple.
https://paulthepoke.com/2016/05/04/trend-update-mount-of-olives-passover-2016/