Can you imagine the Israelis even trying to build a temple next to Al-Asqa right now? And why would they want to build their temple next door to a Muslim mosque squating on holy ground?
I've also read that the true location of the temple is far enough away from the muslim structures that a new temple with the same dimensions as the old could be built on the mount.According to a zealot group called the "Temple Mount Faithful"
"The Temple Mount can never be consecrated to the Name of G-d without removing these pagan shrines. "
So while it's possible, its not the objective of the zealots to do this.
-Eric