Here’s a copy of the analysis written by the Office of Legal Counsel in the twilight days of Nixon’s presidency.
And while it doesn’t make any claim that the Constitution makes a specific exception to the President’s power to pardon that he can’t pardon himself, it acknowledges a commonly accepted legal fundamental that “no one may be a judge in his own case”.
If Trump chose to pardon himself, and did so with the claim that the Constitution doesn’t say he can’t, the matter would probably wind up with the Supreme Court. And the Supreme Court would probably be reluctant to grant this kind of power to the President, as it would apply to all future Presidents (including Democrats!).
But even if the SCOTUS upheld Trump’s right to pardon himself, by pardoning himself, Trump would be essentially admitting that he’s guilty of something requiring a pardon. And it wouldn’t stop him from being impeached and removed from office. It would, however, stop him from being arrested and charged in a civil court, and he wouldn’t go to jail. However, his admission of guilt would still destroy him politically and commercially, and he’d live out his days as a lonely old man, shunned by Americans much in the manner Bill Cosby has been. It wouldn’t be a smart move, unless jail was a certain alternative.