  We report site-selective $^{63}$Cu NMR studies of the vortex core states of
an overdoped Tl$_{2}$Ba$_{2}$CuO$_{6+\delta}$ with $T_{c}$ = 85 K. We observed
a relatively high density of low-energy quasi-particle excitations at the
vortex cores in a magnetic field of 7.4847 T along the c axis, in contrast to
YBa$_{2}$Cu$_{3}$O$_{7-\delta}$.
