The polarization of prompt J/psi, at the Fermilab Tevatron is calculated wi
thin the nonrelativistic QCD factorization framework. The contribution from
radiative decays of P-wave charmonium states decreases, but does not elimi
nate, the transverse polarization at large transverse momentum. The angular
distribution parameter alpha for leptonic decays of the J/psi is predicted
to increase from near 0 at p(T)=5 GeV to about 0.5 at p(T)=20 GeV. The pre
diction is consistent with measurements by the CDF Collaboration at interme
diate values of p(T), but disagrees by about 3 standard deviations at the l
argest values of p(T) measured.