But the court which is without the temple leave out, and measure it not; for it is given unto the Gentiles: and the holy city shall they tread under foot forty and two months.
Revelation 11:1-2 KJV
It is conceivable the Israelis could build their Third Temple without tearing down the two mosques, but what would be the point? Doing so would clearly be going against G-d's will.
Note well that reports such as the one above NEVER seem to call this place "the Noble Sanctuary" or any other name which acknowledges that it legally belongs to the Palestinians. And that is consistent with the Israeli intention to make it theirs.
To me this confirms that the two mosques WILL be torn down:
As far as I am concerned, any claims to the contrary are lies.