  Based on 368.5 pb$^{-1}$ of $e^+e^-$ collision data collected at
center-of-mass energies 4.914 and 4.946 GeV by the BESIII detector, the $e^+e^-
\to \phi\chi_{c1}(3872)$ process is searched for the first time. No significant
signal is observed and the upper limits at the 90\% confidence level on the
product of the Born cross section $\sigma(e^+e^- \to \phi\chi_{c1}(3872))$ and
the branching fraction $\mathcal{B}[\chi_{c1}(3872)\to\pi^+\pi^- J/\psi]$ at
4.914 and 4.946 GeV are set to be 0.85 and 0.96 pb, respectively. These
measurements provide useful information for the production of the
$\chi_{c1}(3872)$ at $e^+e^-$ collider and deepen our understanding about the
nature of this particle.
