The answer is that yes, the President’s power of pardon allows this. However, the power of pardon extends only to *criminal* activities. Impeachment is a political, not a criminal process. Hence, even if a President pardoned himself for the bill of indictment in an impeachment, it would only affect criminal penalties for the underlying act, not removal from office.
Remember, Andrew Johnson, Richard Nixon, and Bill Clinton were not subject to criminal sanction under their bills of impeachment — only removal from office. Any jail time would have had to result from a future criminal trial, regardless of whether the Senate convicted them on the bill of impeachment.
Note that no President, even Nixon, has ever issued a pardon to himself. But that is because they do not wish to admit wrongdoing, and are willing to bet that no criminal charges will follow their exit from office, not because there is any Constitutional bar from their doing so.
Nixon was not impeached.