We compare predicted Type Ib/c supernova (SN) rates with the observed long-duration gamma-ray burst (GRB) rates both locally and as a function of redshift. To do that, we assume different star formation histories in galaxies of different morphological type: ellipticals, spirals and irregulars. In particular, the predicted Type Ib/c SN rate (SNRIb/c) is always higher than the GRB rate, as expected, suggesting that only a small fraction (0.1-1%) of them become GRBs. We predict a ratio between the cosmic GRB rate and the cosmic SNRIb/c in the range 10-2 - 10-3, in agreement with previous estimates. Finally, due to the high star formation in spheroids at high redshift, we predict more GRBs at high redshift than previous estimates, a prediction which awaits to be proven by future observations by GLAST. Based on our studies and on the current LAT performance, an estimate of the detection possibility of this high-redshift burst population is finally presented.
