  A search for the decay $h_c\rightarrow\pi^0J/\psi$ is performed using a
sample of $h_c$ produced in the reaction $e^+e^-\rightarrow\pi^+\pi^-h_c$. The
data samples were collected with the BESIII detector at center-of-mass energies
between 4.189 and 4.437 GeV, corresponding to a total integrated luminosity of
11 fb$^{-1}$. No significant signal is observed. Upper limits on the branching
ratio
$\mathcal{B}(h_c\rightarrow\pi^0J/\psi)/\mathcal{B}(h_c\rightarrow\gamma\eta_c\rightarrow\gamma
K^+K^-\pi^0)$ and on the branching fraction
$\mathcal{B}(h_c\rightarrow\pi^0J/\psi)$ are determined to be
$7.5\times10^{-2}$ and $4.7\times10^{-4}$ at $90\%$ confidence level,
respectively. The latter is derived from the former using the measured
branching fraction of the normalization channel. This is the first
determination of the upper limit of the decay $h_c\rightarrow\pi^0J/\psi$.
