My reading of the U.S. Constitution is that POTUS runs the entire executive branch, which includes the Department of Justice. So he can order them to investigate whatever he wants. If it is something inappropriate, there might be a political price to pay, but nothing is off limits.
The article points that out.
“To paraphrase Justice Scalia, the term “the Executive Power” does not mean some of that power or most of it; it means all of it. “