Observations imply that gamma-ray bursts (GRBs) are produced by the dissipation of the kinetic energy of a highly relativistic fireball. Photo-meson interactions of protons with gamma-rays within the fireball dissipation region are expected to convert a significant fraction of fireball energy to > 10(14) eV neutrinos. We show that the fraction of fireball energy converted in this process to high energy neutrinos is not very sensitive to uncertainties in fireball model parameters, such as the expansion Lorentz factor and characteristic variability time. This is due in part to the constraints imposed on fireball parameters by observed GRB characteristics, and, more important, to the fact that for parameter values for which the photo-meson optical depth is high (implying high proton energy loss to pion production) neutrino production is suppressed by pion and muon synchrotron losses and by competition between two-photon annihilation and photo-meson interactions. The neutrino flux is therefore expected to be correlated mainly with the observed gamma-ray flux.