The Antarctic Impulsive Transient Antenna (ANITA) collaboration has reported a total of three neutrino candidates from the experiment's first three flights. One of these was the lone candidate in a search for Askaryan radio emission, and the others can be interpreted as tau-neutrinos, with important caveats. Among a variety of explanations for these events, they may be produced by astrophysical transients with various characteristic timescales. We test the hypothesis that these events are astrophysical in origin by searching for IceCube counterparts. Using seven years of IceCube data from 2011 through 2018, we search for neutrino point sources using integrated, triggered, and untriggered approaches, and account for the substantial uncertainty in the directional reconstruction of the ANITA events. Due to its large livetime and effective area over many orders of magnitude in energy, IceCube is well suited to test the astrophysical origin of the ANITA events.