A search is reported for near-threshold structures in the $J/\psi J/\psi$ invariant mass spectrum produced in proton-proton collisions at $\sqrt{s}=13\mathrm{TeV}$ from data collected by the CMS experiment, corresponding to an integrated luminosity of $135{\mathrm{fb}}^{-1}$. Three structures are found, and a model with quantum interference among these structures provides a good description of the data. A new structure is observed with a local significance above 5 standard deviations at a mass of ${6638}_{-38}^{+43}{\left(\mathrm{stat}\right)}_{-31}^{+16}\left(\mathrm{syst}\right)\mathrm{MeV}$. Another structure with even higher significance is found at a mass of ${6847}_{-28}^{+44}{\left(\mathrm{stat}\right)}_{-20}^{+48}\left(\mathrm{syst}\right)\mathrm{MeV}$, which is consistent with the $X\left(6900\right)$ resonance reported by the LHCb experiment and confirmed by the ATLAS experiment. Evidence for another new structure, with a local significance of 4.7 standard deviations, is found at a mass of ${7134}_{-25}^{+48}{\left(\mathrm{stat}\right)}_{-15}^{+41}\left(\mathrm{syst}\right)\mathrm{MeV}$. Results are also reported for a model without interference, which does not fit the data as well and shows mass shifts up to 150 MeV relative to the model with interference.