The Delilah hitmaker was friends with both singers back in his music heyday, and he covered a number of songs previously made famous by Sinatra, including My Way and Fly Me To The Moon, with the blessing of the Rat Pack star.
However, Presley was less enthusiastic about the Welshman's takes on the classic tracks and warned Jones to stay away from the iconic singer's music.
Jones tells British radio station Magic 105.4, ''I knew Frank Sinatra and Elvis Presley very well. I did an album of Frank Sinatra type things and Elvis listened and said, 'Tom I heard that thing' and I said... 'Yeah and?' Elvis said, 'We leave that to Frank Sinatra, we don't go there'.''
However, Jones insists Sinatra enjoyed his crooner style and was concerned when he started releasing more rock-orientated music.
He continues, "When I do something a little more rocky, Frank would say: 'Tom, when I go and they (management) ask me who could replace me, I say you! So don't go making records like that!'.''