In the beginning of Elvis & Nixon, Michael Shannon doesn't seem much like the King. Gaunt, with his skull pushing at the underside of his sunken face, he speaks softly and moves slowly. But he also has a towering physical presence and homely charm, and soon this odd mix of Harry Dean Stanton and Sylvester Stallone comes to channel Elvis, rather than impersonate him. Jerry Schilling, a member of Elvis's entourage, has said it "may be the best Elvis of all time".

Schilling, played in the film by Alex Pettyfer, accompanied Elvis to the White House on the morning of 21 December 1970, to ask President Nixon to make him a federal agent-at-large. (No one seems to recall why; the film suggests a combination of famous-person boredom and twisted civic duty in the face of an America "sliding into anarchy".) This was before Nixon set up tape recorders in the Oval Office, and all that survives of the talk between the then two most famous white men on the planet is one photo.

In the movie, the imagined conversation is a pleasure to watch, thanks to Shannon and Kevin Spacey, who has just the right (tiny) amount of jowly growl, the fake smiles and cuss-word grumpiness of Nixon. The journey there is not so dramatic — the King and crew try to get the meeting; Nixon nixes it before White House staff fix it so that Elvis enters the building — but it is fun. There are many laughs, including a couple of great Elvis jokes, and Colin Hanks is very good as a stuffy Washington wonk. 

But this is Shannon's show, a label that could be applied to all the films in which he's appeared in over the last five years. One day, he'll be brilliant in a brilliant film and everyone will know who he is.

Elvis and Nixon is out on 24 June