The spatial and temporal coincidence of gamma-ray burst (GRB) 980425 and supernova (SN) 1998bw has prompted speculation that there exists a subclass of GRBs produced by SNe (``S-GRBs’'). A physical model motivated by radio observations lead us to propose the following characteristics of S-GRBs: (1) prompt radio emission and an implied high brightness temperature close to the inverse Compton limit, (2) high expansion velocity (>åisebox-0.5ex 50,000 km s$^-1$) of the optical photosphere as derived from lines widths and energy release larger than usual, (3) no long-lived X-ray afterglow, and (4) a single-pulse GRB profile. Radio studies of previous SNe show that only (but not all) Type Ib and Ic SNe potentially satisfy the first condition. We investigate the proposed associations of GRBs and SNe within the context of these proposed criteria and suggest that i̊sebox-0.5ex 1% of GRBs detected by BATSE may be members of this subclass.