  The decay $J/\psi \to \omega p \bar p$ is studied using a $5.8 \times 10^7$
$J/\psi$ event sample accumulated with the BES II detector at the Beijing
electron-positron collider. The decay branching fraction is measured to be
$B(J/\psi \to \omega p \bar p)=(9.8\pm 0.3\pm 1.4)\times 10^{-4}$. No
significant enhancement near the $p\bar p$ mass threshold is observed, and an
upper limit of $B(J/\psi \to \omega X(1860))B(X(1860)\to p\bar p)$$ $$< 1.5
\times 10^{-5}$ is determined at the 95% confidence level, where X(1860)
designates the near-threshold enhancement seen in the $p\bar p$ mass spectrum
in $J/\psi \to \gamma p \bar p$ decays.
