The Cu-63 Knight shifts have been measured for the heavily doped high-T(c) superconductors TlSr2CaCu2O7-y, with T(c) = 64 and 45 K. The spin part of the shift, K(s), is almost T-independent in the normal state and increases with decreasing T(c). From the analysis of the anisotropy in K(s) in both compounds, the transferred hyperfine field from the nearest-neighbor Cu spins has been deduced to be larger than that in YBa2Cu3O7, similar to the results for Tl2Ba2CuO6+y. A systematic correlation has been found in the heavily doped superconductors that the spin susceptibility and the nuclear quadrupole frequency increase with decreasing T(c), while 1/T1T is suppressed, as reported previously.