An amplitude analysis of B0 -> J=psi phi K0S 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=psi K0S system, named T theta psi s1o4000 thorn 0, is seen, with its mass and width measured to be 3991 thorn 12 thorn 9 -10 -17 MeV=c2 and 105 thorn 29 thorn 17 -25 -23 MeV, respectively, where the first uncertainty is statistical and the second systematic. The T theta psi s1 o4000 thorn 0 state is likely to be the isospin partner of the T theta psi s1 o4000 thorn thorn state, previously observed in the J=psi K thorn system of the B thorn -> J=psi phi K thorn decay. When isospin symmetry for the charged and neutral T theta psi s1 o4000 thorn states is assumed, the signal significance increases to 5.4 standard deviations.