  The Large High Altitude Air Shower Observatory (LHAASO) observed a dozen of
gamma-ray sources with significant emission above 100 TeV, which may be strong
candidates of PeVatrons. Neutrino observations are crucial to diagnose whether
the gamma-ray radiative process is hadronic or leptonic. We use the Bayesian
method to analyze the ten-year (2008-2018) IceCube data, and hence constrain
the hadronic gamma-ray emission in the LHAASO sources. The present neutrino
data show that the hadronic gamma-ray flux from Crab Nebula is lower than the
observed gamma-ray flux at the 90% C.L. and contributes less than 86%, which
disfavors the hadronic origin of the gamma-rays below tens of TeV. For the
other LHAASO sources, the present neutrino observations cannot put useful
constraints on the gamma-ray radiative process. We consider the uncertainty of
the source extension: the upper limits on hadronic gamma-ray flux tend to
increase with the extension; and some sources, i.e., LHAASO J2032+4102, LHAASO
J1929+1745, and LHAASO J1908+0621, show relatively high statistical
significance of neutrino signals if the extension is <0.6 deg. We finally
estimate the future observational results of LHAASO sources by the proposed
neutrino telescopes. If the LHAASO-observed PeV gamma-rays are of hadronic
origin, Crab Nebula may be detected at >100 TeV at 3-sigma C.L. within 20 years
by a neutrino detector with the effective area 30 times that of IceCube.
