The pardon power is defined in Article II, Section 2, Clause 3 of the Constitution. The president has the power to “grant reprieves and pardons for offenses against the United States, except in cases of impeachment.”
The Supreme Court has never ruled on the scope of the president’s pardon power. One argument supporting the notion that the president could issue their own pardon is that the only express limitation on the pardon power is in cases of impeachment. That could lead to the conclusion that the framers of the Constitution didn’t intend limitations on the pardon power beyond impeachment.
An argument against the idea that a president could issue their own pardon could be made around the word “grant.” When something is granted, the grantor is giving a benefit to the grantee. That could imply that a pardon is a two-person transaction where the president is granting another a pardon.
A president’s pardon power only applies to federal crimes. So, a pardon cannot protect someone from criminal charges that may be filed by a state.