Knight shift of Cu-63 and nuclear spin lattice relaxation time T1 of Cu-63 and Tl-205 have been measured in Tl2Ba2CuO6+y with T(c) = 72 K, 40 K and 0 K. In normal state, the spin part of the Cu-63 shift is T-independent and increases with decreasing T(c). 1/T1 of Cu-63 and Tl-205 obey a T1T = constant law over a wide T-range from about 150 K close to T(c). Above about 150 K and near T(c), (1/T1T) begins to decrease. The T-linear dependence of the resistivity observed in 72 K superconductors changes into a T2 dependence in a non-superconducting metal in which the T1T = constant law holds up to around 200 K. Analysis of the anisotropy of the relaxation rate, 63R = (1/T1)ab(1/T1)c reveals that the antiferromagnetic (AF) spin fluctuations among nearest neighbor Cu sites, which survies still in the compound with T(c) = 72 K, becomes less distinct with decreasing T(c) and disappears in a non-superconducting sample. These results point out that there may be an intimate interrelation between the occurrence of superconductivity and the presence of the AF spin correlation, although the magnetic coherence length, xi-M is supposed to be as small as xi-M/a less-than-or-equal-to approximately 1. Experimentally, it is shown that the T-linear dependent resistivity is relevant to the presence of the AF spin correlation. In the superconducting state, the relaxation behavior is found to possess a universal feature in most of the high-T(c) oxides, indicating a common relaxation mechanism to work even in heavily-doped superconductors exceeding the maximum of T(c).