The spatial and temporal coincidence between the gamma-ray burst (GRB) 980425 and supernova (SN) 1998bw has prompted speculation that there exists a class of GRBs produced by SNe (``S-GRBs''). Robust arguments for the existence of a relativistic shock have been presented on the basis of radio observations. A physical model based on the radio observations lead us to propose the following characteristics of supernovae GRBs (S-GRBs): 1) prompt radio emission and implied brightness temperature near or below the inverse Compton limit, 2) high expansion velocity 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 (SP) GRB profile. Radio studies of previous SNe show that only type Ib and Ic potentially satisfy the first condition. Accordingly we have investigated proposed associations of GRBs and SNe finding no convincing evidence (mainly to paucity of data) to confirm any single connection of a SN with a GRB. If there is a more constraining physical basis for the burst time-history of S-GRBs beyond that of the SP requirement, we suggest the 1% of light curves in the BATSE catalogue similar to that of GRB 980425 may constitute the subclass. Future optical follow-up of bursts with similar profiles should confirm if such GRBs originate from some fraction of SN type Ib/Ic. ; Comment: 11 pages of LaTeX with 1 figure. Submitted to the Astrophysical Journal Letters