Jan & Dean's version of "Memphis" first appeared on the Surf City and Other Swingin' Cities album which came out in 1963. (The track was later reused for The Little Old Lady from Pasadena album in 1964.)
Johnny Rivers at the Whisky a Go Go was released in February 1964, so that was after J&D's version was released on SCAOSC. However, it is very possible that J&D's version could have been influenced by Johnny's since Johnny had most likely been performing it live for quite awhile prior to it's appearance on the album. Legend has it that Johnny's arrangement of the song was "stolen" from an unreleased version of the song by Elvis Presley that a member of Elvis' team had played for Johnny. As a result, Elvis allegedly hated Johnny for the rest of his life.
Lou Adler produced the album which is probably where J&D's endorsement on the cover came from. In addition, Johnny's label, Imperial Records, had just been purchased by Liberty Records in 1963, so it could have been cross-promotional as well.