Does Trump have the power to pardon himself?

This question arises from a Trump tweet last Saturday when Trump said this: ‘While all agree the U. S. President has the complete power to pardon, why think of that when only crime so far is LEAKS against us. FAKE NEWS.”
Why think of that indeed, but can the President pardon himself or is this too close to Nixon’s ‘when the President does it that means it isn’t illegal’.