  We report a measurement of the observed cross sections of the inclusive
$J/\psi$ production in $e^+e^-\rightarrow {J}/\psi{\rm X}$ based on 3.21
fb$^{-1}$ of data accumulated at energies from 3.645 to 3.891 GeV with the
BESIII detector operated at the BEPCII collider. The energy-dependent lineshape
obtained from the measured cross sections cannot be well described by two
Breit-Wigner (BW) amplitudes of the expected decays $\psi(3686)\rightarrow
{J}/\psi{\rm X}$ and $\psi(3770)\rightarrow {J}/\psi{\rm X}$. Instead it can be
better described with three BW amplitudes of the decays $\psi(3686)\rightarrow
{J}/\psi{\rm X}$, $R(3760)\rightarrow {J}/\psi{\rm X}$ and $R(3790)\rightarrow
{J}/\psi{\rm X}$ with two distinct structures referred to as $R(3760)$ and
$R(3790)$. Under this assumption, we extracted their masses, total widths, and
the product of the leptonic width and decay branching fractions to be
$M_{R(3760)}= {3761.7\pm 2.2 \pm 1.2}$ MeV/$c^2$, $\Gamma^{\rm tot}_{R(3760)}=
{6.7\pm 11.1 \pm 1.1}$ MeV, $\Gamma^{ee}_{R(3760)}\mathcal B[R(3760)\rightarrow
{J}/\psi {\rm X}]=(4.0\pm 4.3\pm 1.2)$ eV, $M_{R(3790)} = {3784.7\pm 5.7 \pm
1.6}$ MeV/$c^2$, $\Gamma^{\rm tot}_{R(3790)} = {31.6 \pm 11.9 \pm 3.2}$ MeV,
$\Gamma^{ee}_{R(3790)}\mathcal B[R(3790)\rightarrow {J}/\psi {\rm X}]=(18.1\pm
10.3\pm 4.7)$ eV, where the first uncertainties are statistical and second
systematic.
