Article I, Section 9, Clause 2: "The Privilege of the Writ of Habeas Corpus shall not be suspended, unless when in Cases of Rebellion or Invasion the public Safety may require it." I don't see where it says it's a legislative power or that only Congress can suspend it.
In Ex parte Milligan, 71 US 2 (1866), the Supreme Court held that the denial of habeas corpus to Milligan was in violation of the law.
It ruled that denial of habeas corpus was illegal in areas where the federal courts operated openly. It did not say that habeas corpus could not be suspended.
The fact that it is in Article I of the Constitution is a clue, and in context, the prescription is clearly directed to Congress (as opposed to Section 10 which is clearly directed to the States). And, the fact that Congress passed a measure enabling Lincoln's action is another clue. The Republic is suspended when and where the president successfully asserts the unilateral power to suspend habeas corpus.
It ruled that denial of habeas corpus was illegal in areas where the federal courts operated openly. It did not say that habeas corpus could not be suspended.
My post said basically the same - that Milligan's (but not all) denial of habeas petitions was unconstitutional.