  IceCube has detected many TeV--PeV neutrinos, but their astrophysical origins
remain largely unknown. Motivated by the observed late-time X-ray/optical bumps
in some gamma-ray bursts (GRBs), we examine the correlation between IceCube
neutrinos and GRBs allowing delayed neutrinos $\sim$days after the prompt gamma
rays. Although we have not found any definitive correlation, up to $\sim$10\%
of the events observed so far at IceCube may have been neutrinos produced by
the late-time GRB activities at $\sim$1 day. Assuming a connection between some
IceCube events and the late GRB bumps, we show in a model-independent way that
GRB sites capable of producing late $\sim$PeV neutrinos should be
nonrelativistic or mildly relativistic. We estimate the diffuse neutrino flux
from such sources and find that they can possibly account for a few IceCube
events. Future observations of high-energy neutrinos and late-time GRB
afterglows can further test the above proposed connection.
