  Using 9.9 fb$^{-1}$ of $e^+e^-$ collision data collected by the BESIII
detector at center-of-mass energies between 4.15 and 4.30 GeV, we search for
the processes $e^+e^-\to\gamma X(3872)$ with $X(3872)\rightarrow\pi^0\chi_{c0}$
and $X(3872)\rightarrow\pi\pi\chi_{c0}$. Depending on the fitting model, the
statistical significance for $X(3872)\to\pi^0\chi_{c0}$ ranges from 1.3$\sigma$
to 2.8$\sigma$. We set upper limits (at 90\% C.L.) of
$\frac{\mathcal{B}(X(3872)\rightarrow\pi^0\chi_{c0})}{\mathcal{B}(X(3872)\to\pi^+\pi^-J/\psi)}<3.6$,
$\frac{\mathcal{B}(X(3872)\rightarrow\pi^+\pi^-\chi_{c0})}{\mathcal{B}(X(3872)\to\pi^+\pi^-J/\psi)}<0.68$,
and
$\frac{\mathcal{B}(X(3872)\rightarrow\pi^0\pi^0\chi_{c0})}{\mathcal{B}(X(3872)\to\pi^+\pi^-J/\psi)}<1.7$.
Combined with the BESIII measurement of $X(3872)\to\pi^0\chi_{c1}$, we also set
an upper limit of
$\frac{\mathcal{B}(X(3872)\rightarrow\pi^0\chi_{c0})}{\mathcal{B}(X(3872)\to\pi^0\chi_{c1})}<4.4$.
