The funnyman serenaded his wife with a version of Presley's track One Night with You during the pre-taped stand-up performance for America's Showtime network.
However, the programme had to be re-edited to omit the sequence after Clay failed to win permission to include the song.
He tells New York Post gossip column Page Six, "I've always been tight with (Presley's associate) Jerry Schilling and the 'Memphis Mafia' thing. But they don't run the estate anymore. They wanted a lot of money for it (the song). Money is the root of all evil."
The comic's show, titled Indestructible, will air in America on 31 December (12).