  We present a study of the process $e^{+}e^{-}\to\eta\phi$ using data samples
collected with the BESIII detector corresponding to an integrated luminosity of
15.03 fb$^{-1}$ at 23 center-of-mass energies from 3.773 to 4.600 GeV. The Born
cross sections are measured at each energy and a coherent fit to cross-section
lineshape is performed using a Breit-Wigner parametrization to search for
charmonium-like vector states. No significant signals of the $Y(4230)$ and
$Y(4360)$ resonances are observed.
