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 $M_{R(3760)}= {3761.7pm 2.2 pm 1.2}$ MeV/$c^2$, $Gamma^{rm tot}_{R(3760)}= {6.7pm 11.1 pm 1.1}$ MeV, $Gamma^{ee}_{R(3760)}mathcal B[R(3760)rightarrow {J}/psi {rm X}]=(4.0pm 4.3pm 1.2)$ eV, $M_{R(3790)} = {3784.7pm 5.7 pm 1.6}$ MeV/$c^2$, $Gamma^{rm tot}_{R(3790)} = {31.6 pm 11.9 pm 3.2}$ MeV, $Gamma^{ee}_{R(3790)}mathcal B[R(3790)rightarrow {J}/psi {rm X}]=(18.1pm 10.3pm 4.7)$ eV, where the first uncertainties are statistical and second systematic.
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