Blazar jets are extreme environments, in which relativistic proton interactions with an ultraviolet photon field could give rise to photopion production. High-confidence associations of individual high-energy neutrinos with blazar flares could be achieved via spatially and temporally coincident detections. In 2017, the track-like, extremely high-energy neutrino event IC 170922A was found to coincide with increased γ-ray emission from the blazar TXS 0506+056, leading to the identification of the most promising neutrino point-source candidate so far. We calculate the expected number of neutrino events that can be detected with IceCube, based on a broadband parameterization of bright short-term blazar flares that were observed in the first 6.5 yr of Fermi/Large Area Telescope observations. We find that the integrated keV-to-GeV fluence of most individual blazar flares is far too small to yield a substantial Poisson probability for the detection of one or more neutrinos with IceCube. We show that the sample of potentially detectable high-energy neutrinos from individual blazar flares is rather small. We further show that the blazars 3C 279 and PKS 1510−089 dominate the all-sky neutrino prediction from bright and short-term blazar flares. In the end, we discuss strategies to search for more significant associations in future data unblindings of IceCube and KM3NeT.