  The process $e^+e^-\rightarrow p\overline{p}\pi^0$ has been studied by
analyzing data collected at $\sqrt{s}=3.773$ GeV, at $\sqrt{s}=3.650$ GeV, and
during a $\psi(3770)$ line shape scan with the BESIII detector at the BEPCII
collider. The Born cross section of $p \overline{p} \pi^0$ in the vicinity of
the $\psi(3770)$ is measured and the Born cross section of
$\psi(3770)\rightarrow p \overline{p} \pi^0$ is extracted considering
interference between resonant and continuum production amplitudes. Two
solutions with the same probability and a significance of 1.5$\sigma$ are
found, and the Born cross section of $\psi(3770)\rightarrow p \overline{p}
\pi^0$ is determined to be less than 0.22 pb at 90% confidence level and
$33.8\pm1.8\pm2.1$ pb, respectively. Using the estimated cross section and a
constant decay amplitude approximation, the cross section $\sigma(p\overline{p}
\rightarrow \psi(3770) \pi^0)$ is calculated for the kinematic situation of the
planned $\overline{\text{P}}\text{ANDA}$ experiment. The maximum cross section
corresponding to the two solutions is expected to be less than $0.79$ nb at 90%
confidence level and $122\pm10$ nb at a center of mass energy of 5.26 GeV.
