As my co-blogger Ilya Somin notes below, White House aide Stephen Miller commented today that the Trump Administration is "actively looking at" whether it would be possible to suspend the writ of habeas corpus on the grounds that the country is suffering an "invasion" by illegal immigrants.
My own view is that the White House can look at this question all it wants, but it is ultimately up to Congress whether the writ should be suspended (at least during times of peace). This is clear form the Constitution's text and structure. The suspension clause is in Article I, section 9, as among several enumerated constraints on legislative power. It is an interesting question whether Courts can review a legislative suspension of the writ, but I think it is relatively clear that the Executive cannot do so unilaterally.
I also think that the Supreme Court has fairly consistently operated under this assumption. Consider Hamdi v. Rumsfeld. In that case there was unanimous agreement that Congress had not suspended the writ, and so the question was whether the federal government could detain Hamdi (an American citizen alleged to be an enemy combatant captured in Afghanistan) in the United States without putting him on trial.
