  We present a calculation for the polarizations of $J/\psi$ and $\psi'$
produced in the hadron collisions at the Fermilab Tevatron. Various color octet
channels including ${}^1S_0^{(8)}$, ${}^3P_J^{(8)}$, and ${}^3S_1^{(8)}$ as
well as contributions from $\chi_{cJ}$ decays are considered in the $k_t$
factorization approach. We find that in a rather wide range of the transverse
momenta of $J/\psi$ and $\psi'$, the production rates could be dominated by the
${}^1S_0^{(8)}$ channel, and the predicted polarizations from the
${}^1S_0^{(8)}$ channel and $\chi_{cJ}$ feeddown contributions are roughly
compatible with the preliminary CDF data. This might provide a possible release
from the conflict between the NRQCD collinear parton model calculations and the
CDF data.
