An amplitude analysis of B^{0}→J/ψϕK_{S}^{0} decays is performed using proton-proton collision data, corresponding to an integrated luminosity of 9 fb^{-1}, collected with the LHCb detector at center-of-mass energies of 7, 8, and 13 TeV. Evidence with a significance of 4.0 standard deviations of a structure in the J/ψK_{S}^{0} system, named T_{ψs1}^{θ}(4000)^{0}, is seen, with its mass and width measured to be 3991_{-10}^{+12} _{-17}^{+9} MeV/c^{2} and 105_{-25}^{+29} _{-23}^{+17} MeV, respectively, where the first uncertainty is statistical and the second systematic. The T_{ψs1}^{θ}(4000)^{0} state is likely to be the isospin partner of the T_{ψs1}^{θ}(4000)^{+} state, previously observed in the J/ψK^{+} system of the B^{+}→J/ψϕK^{+} decay. When isospin symmetry for the charged and neutral T_{ψs1}^{θ}(4000) states is assumed, the signal significance increases to 5.4 standard deviations.