  Using a data sample corresponding to an integrated luminosity of 2.93
fb$^{-1}$ collected at a center-of-mass energy of 3.773~GeV with the BESIII
detector at the BEPCII collider, we search for a scalar partner of the
$X(3872)$, denoted as $X(3700)$, via $\psi(3770)\to \gamma\eta\eta'$ and
$\gamma\pi^{+}\pi^{-}J/\psi$ processes. No significant signals are observed and
the upper limits of the product branching fractions $ {\cal
B}(\psi(3770)\to\gamma X(3700))\cdot {\cal B}(X(3700)\to \eta\eta')$ and ${\cal
B}(\psi(3770)\to\gamma X(3700))\cdot {\cal B}(X(3700)\to\pi^{+}\pi^{-}J/\psi)$
are determined at the 90\% confidence level, for the narrow $X(3700)$ with a
mass ranging from 3710 to 3740 MeV/$c^2$, which are from 0.8 to 1.8 $(\times
10^{-5})$ and 0.9 to 3.4 $(\times 10^{-5})$, respectively.
