According to a report from Associated Press, the Trump administration floated the idea of using 100,000 National Guard troops of 11 border and border-adjacent states to round up illegal immigrants. (Spicy denies the existence of such a proposal.)
Wikipedia: Posse comitatus is the common-law or statute law authority of a county sheriff, or other law officer, to conscript any able-bodied man to assist him in keeping the peace or to pursue and arrest a felon, similar to the concept of the "hue and cry." Originally found in English common law, it is generally obsolete; however, it survives in the United States, where it is the law enforcement equivalent of summoning the militia for military purposes.
The Posse Comitatus Act was passed in 1878, in the wake of the deadlocked presidential election of 1876 in the aftermath of the Civil War, during the period of Reconstruction. In a tradeoff, the U.S. government agreed to pull federal troops out of the South, so that they would no longer enforce laws about voting, lynching, and suchlike. That ushered in the era of Ku Klux Klan terror and 75 years of Jim Crow.
The Trump proposal would try to get around that law by asking the governors of 11 southern states to cooperate in using their National Guard troops to assist federal immigration officers -- ICE -- in rounding up and deporting undocumented immigrants. But the proposal wouldn't seek to federalize them; that means the states would have to pay the expense -- and most of the states can't afford it.
Trump could federalize the National Guard in those 11 states, but that would mean the federal government would have to pay for it, and that requires an act of Congress -- which probably won't happen.