In this work, we investigate the likelihood of association between real-time, neutrino alerts with teraelectronvolt to petaelectronvolt energy from IceCube and optical counterparts in the form of core-collapse supernovae (CC SNe). The optical follow-up of IceCube alerts requires two main instrumental capabilities: (1) deep imaging, since 73% of neutrinos would come from CC SNe at redshifts z > 0.3, and (2) a large field of view (FoV), since typical IceCube muon neutrino pointing accuracy is on the order of similar to 1 deg. With Blanco/DECam (gri to 24th magnitude and 2.2 deg diameter FoV), we performed a triggered optical follow-up observation of two IceCube alerts, IC170922A and IC171106A, on six. nights during the three weeks following each alert. For the IC170922A (IC171106A) follow-up observations, we expect that 12.1% (9.5%) of coincident CC SNe at z less than or similar to 0.3 are detectable, and that, on average, 0.23 (0.07) unassociated SNe in the neutrino 90% containment regions also pass our selection criteria. We find two candidate CC SNe that are temporally coincident with the neutrino alerts in the FoV, but none in the 90% containment regions, a result that is statistically consistent with expected rates of background CC SNe for these observations. If CC SNe are the dominant source of teraelectronvolt to petaelectronvolt neutrinos, we would expect an excess of coincident CC SNe to be detectable at the 3 sigma confidence level using DECam observations similar to those of this work for similar to 60 (similar to 200) neutrino alerts with (without) redshift information for all candidates.