  Using 10.1 fb$^{-1}$ of $e^+e^-$ collision data collected by the BESIII
detector with center-of-mass energies between 4.15 GeV and 4.30 GeV, we search
for the decays $X(3872)\to\pi^0\pi^0\chi_{c1,2}$, where the $X(3872)$ is
produced in $e^+e^-\to\gamma X(3872)$. No evidence above $3\sigma$ is found for
either decay. Upper limits at the $90\%$ C.L. on the branching fractions of
$X(3872)\to\pi^0\pi^0\chi_{c1,2}$ normalized to the branching fraction of
$X(3872)\to\pi^+\pi^-J/\psi$ are set to be
$\mathcal{B}(X(3872)\to\pi^0\pi^0\chi_{c1})/\mathcal{B}(X(3872)\to\pi^+\pi^-J/\psi)
< 1.1$ and
$\mathcal{B}(X(3872)\to\pi^0\pi^0\chi_{c2})/\mathcal{B}(X(3872)\to\pi^+\pi^-J/\psi)
< 0.5$, taking into account both statistical and systematic uncertainties.
