The $S$-wave scattering amplitudes of $J/\psi J/\psi$ with quantum numbers $J^{PC} = 0^{++}$ and $2^{++}$ are determined up to 6600\,MeV using lattice QCD calculations at $m_\pi \approx 420$ and 250\,MeV. The ${}^1S_0$ $J/\psi J/\psi$ system exhibits a near-threshold attractive interaction, resulting in a virtual bound state with a binding energy of approximately 30-40\,MeV. In contrast, the ${}^5S_2$ $J/\psi J/\psi$ system exhibits a repulsive interaction near threshold. These behaviors are primarily dominated by the quark rearrangement effect. Most notably, a resonance is observed in the ${}^5S_2$ $J/\psi J/\psi$ channel, with a mass around 6540\,MeV and a width of approximately 540\,MeV. The extracted mass and width are consistent with the $X(6600)$ (or $X(6400)$) observed by the ATLAS and CMS collaborations, and show little dependence on the sea pion mass.