We have measured the temperature dependence of 1/T1 at the O, Cu T1 sites of Tl2Ba2CuOy. 1/T1 at the T1 sites does not follow the Korringa law for over-doped and superconducting specimens (4.2-300K). On the other hand, 1/T1 at the O and Cu sites appeared to be the Korringa type (T1xT = const) up to 200K for the over-doped specimen. From a comparison of the magnitude of 1/T1 at the O and Cu sites, we concluded that the antiferromagnetic spin fluctuation are playing essential role to the spin-dynamics at Cu site even in the over-doped state.