  Nuclear magnetic resonance measurements were performed on CeCu$_{2}$Si$_{2}$
in the presence of a magnetic field close to the upper critical field $\mu_{0}
H_{\rm c2}$ in order to investigate its superconducting (SC) properties near
pair-breaking fields. In lower fields, the Knight shift and nuclear
spin-lattice relaxation rate divided by temperature $1/T_1T$ abruptly decreased
below the SC transition temperature $T_{\rm c}(H)$, a phenomenon understood
within the framework of conventional spin-singlet superconductivity. In
contrast, $1/T_1T$ was enhanced just below $T_{\rm c}(H)$ and exhibited a broad
maximum when magnetic fields close to $\mu_0H_{\rm c2}(0)$ were applied
parallel or perpendicular to the $c$ axis, although the Knight shift decreased
just below $T_{\rm c}(H)$. This enhancement of $1/T_1T$, which was recently
observed in the organic superconductor $\kappa$-(BEDT-TTF)$_{2}$Cu(NCS)$_{2}$,
suggests the presence of high-density Andreev bound states in the inhomogeneous
SC region, a hallmark of the Fulde-Ferrell-Larkin-Ovchinnikov phase.
