A search is made for potential ccc[over ¯]c[over ¯] tetraquarks decaying into a pair of charmonium states in the four muon final state using proton-proton collision data at sqrt[s]=13 TeV, corresponding to an integrated luminosity of 140 fb^{-1} recorded by the ATLAS experiment at LHC. Two decay channels, J/ψ+J/ψ→4μ and J/ψ+ψ(2S)→4μ, are studied. Backgrounds are estimated based on a hybrid approach involving Monte Carlo simulations and data-driven methods. Statistically significant excesses with respect to backgrounds dominated by the single parton scattering are seen in the di-J/ψ channel consistent with a narrow resonance at 6.9 GeV and a broader structure at lower mass. A statistically significant excess is also seen in the J/ψ+ψ(2S) channel. The fitted masses and decay widths of the structures are reported.