There is a semi significant effort to rebuild the third Beit Hamikdash, but several obvious problems are: One, we don't know the exact location and probably won't confirm it for some time due to the prevention of archaeological work and destruction of artifacts by the Arabs there. Second, most probably it is under the dome, so then that's off limits at least for now.
However, it says in the Torah that G-d commended Israel to worship and sacrifice in the "place that G-d will choose" for His name to be honored. Shilo was such a place and still is the place that for 369 years the kodosh kodoshim was present. There is strong evidence we know the exact spot there. The mishkan was always a temporary spot till the Temple was built. Why not rebuild the mishkan and wait for the building of the third temple later as a final resting place?