The US Constitution is not as silent on the issue as you think.
Article II, Section 2, Clause 1 of the US Constitution states:
That already limits the presidential pardon to only those offenses against the United States. Meaning the President may not pardon himself, or anyone else, involved in a private civil or State criminal matter. The President's pardon authority extends only to federal charges.
With regard to federal charges, however, the Supreme Court in
Ex parte Garland, 71 U.S. (4 Wall.) 333 (1866) held in a 5-4 decision that the President's power to grant pardons was "unlimited," except for impeachments.
In
Murphy v. Ford, 390 F. Supp. 1372 (W.D. Mich. 1975), the Federal District Court in Michigan rejected a suit for a declaratory judgment that President Ford's unconditional pardon of Nixon was unconstitutional. The court found that the President had the constitutional power to grant a pre-indictment pardon, citing
Ex parte Garland in its support.
The President may even pardon individuals before they are charged with a crime.
The only valid argument against a President pardoning themselves is in
Federalist #10, where Madison states that "No man is allowed to be a judge in his own cause, because his interest would certainly bias his judgment, and, not improbably, corrupt his integrity."
However, beyond that statement in Federalist #10, there seems to be no compelling legal argument against the presidential pardon including the President. Politically speaking it would be a disaster, and would completely undermine everything the President attempted to do thereafter.
The solution to prevent a President from pardoning themselves, of course, is to bring up Articles of Impeachment in the House. Failing an impeachment by the House, the President does indeed have the constitutional authority to pardon any federal offense, real or imagined, before or after-the-fact.