  Relativistic corrections to the color-singlet J/\psi hadroproduction at the
Tevatron and LHC are calculated up to O(v^2) in nonrelativistic QCD (NRQCD).
The short distance coefficients are obtained by matching full QCD with NRQCD
results for the subprocess g+g\to J/\psi+g. The long distance matrix elements
are extracted from observed J/\psi hadronic and leptonic decay widths up to
O}(v^2). Using the CTEQ6 parton distribution functions, we calculate the LO
production cross sections and relativistic corrections for the process
p+\bar{p}(p)\to J/\psi+X at the Tevatron and LHC. We find that the enhancement
of O(v^2) relativistic corrections to the cross sections over a wide range of
large transverse momentum p_t is negligible, only at a level of about 1 %. This
tiny effect is due to the smallness of the correction to short distance
coefficients and the suppression from long distance matrix elements. These
results indicate that relativistic corrections can not help to resolve the
large discrepancy between leading order prediction and experimental data for
J/\psi production at the Tevatron.
