Robert Naunton was the son of Henry Naunton of Alderton, Suffolk, and Elizabeth Asheby of Hornsby, Leicestershire. According to Schreiber, the Nauntons were 'established members of the county gentry and had been so for well over two centuries'. Robert Naunton's grandfather, William Naunton, was trained as a lawyer and married Elizabeth Wingfield, the daughter Sir Anthony Wingfield, a trusted servant of Henry VIII.