  The site-selective nuclear spin-lattice relaxation rate T1^{-1} is
theoretically studied inside a vortex core in a chiral p-wave superconductor
within the framework of the quasiclassical theory of superconductivity. It is
found that T1^{-1} at the vortex center depends on the sense of the chirality
relative to the sense of the magnetic field. Our numerical result shows a
characteristic difference in T1^{-1} between the two chiral states, k_x + i k_y
and k_x - i k_y under the magnetic field.
